The Lagrangian of a relativistic (quantum) field theory for a certain set of (scalar, vector, spin $1/2$) fields takes the general form (see also https://physics.stackexchange.com/q/846005) $$ \mathcal{L} = \sum\limits_{i=1}^n c_i O_i, \tag{1} \label{1} $$
where \eqref{1} contains all (linear independent) local (hermitean) operators $O_i$ (constructed in terms of products of the basic fields and their derivatives) respecting the necessary space-time symmetries (the action integral $\int\! d^4 x \, \mathcal{L}$ must be invariant under $\mathcal{P}_+^\uparrow$) and possibly also discrete symmetries (like parity $ P$ and $CP$, etc.) and/or internal symmetries (like local or global gauge symmetries). Demanding also renormalizability, the $O_i$ in \eqref{1} are further restricted to the subset of operators with dimension $\le 4$.
In general, the coefficients $c_i$ associated with the operators $O_i$ in \eqref{1} do not correspond to (directly) observable parameters of the theory. However, they can be related to observable quantities by the following procedure: Choose $n$ suitable (independent) experimentally determined (physical) values $a_{i, \rm ph}$ (like masses, cross sections, decay widths, etc.) and compute the corresponding theoretical expressions $a_i(c_1, \ldots, c_n)$ using the (quantum) field theory defined by the Lagrangian \eqref{1}, arriving at the relations $$ a_{i, \rm ph} =a_i(c_1, \ldots, c_n), \quad i=1, \ldots,n. \tag{2} \label{2} $$ In principle, the set of equations in \eqref{2} can be inverted, expressing the original parameters $c_i$ in terms of the measured ones, $$ c_i=c_i(a_{1, \rm ph}, \ldots, a_{n, \rm ph}), \quad i=1,\ldots, n. \tag{3} \label{3} $$ Note that this approach does not necessarily require a quantized field theory but can equally well be applied to a classical model.
In the case of a quantum field theory, one encounters an additional technical problem. The occurrence of ultraviolet divergences generated by large momenta in loop integrals requires the regularization of the theory by introducing a momentum cut-off parameter $\Lambda$. As a consequence, the theoretical expression for the observable $a_i$ is modified to $$ a_i(c_1, \ldots, c_n, \Lambda). \tag{4} \label{4} $$ As \eqref{4} is supposed to reproduce the finite measured values $a_{i, \rm ph}$ in the limit $\Lambda \to \infty$, the coefficients $c_i$ have to be tuned accordingly, $$ c_i \to c_i(\Lambda), \tag{5} \label{5} $$ such that $$ a_{i, \rm ph} = \lim\limits_{\Lambda \to \infty} a_i(c_1(\Lambda), \ldots, c_n(\Lambda), \Lambda) \tag{6} \label{6} $$ holds. In contrast to the physical values $a_{i, \rm ph}$, the coefficients $c_i(\Lambda)$ (being, in general, even divergent in the limit $\Lambda \to \infty$) cannot be regarded as observable parameters of the theory.
As a simple illustration of these somewhat abstract considerations, let us consider the renormalizable relativistic quantum field of a single real scalar field $\phi$ obeying the additional discrete symmetry $\phi \to -\phi$. The associated Lagrangian reads $$ \mathcal{L}= c_0 (\partial_\mu \phi) (\partial^\mu \phi)+ c_1 \phi^2+c_2 \phi^4, \tag{7} \label{7} $$ where the (redundant) parameter $c_0$ can be eliminated by the field rescaling $\phi \to \phi/{\sqrt{2 c_0}}$. With the simultaneous renamings $c_1= -c_0 m^2$ and $c_2= - c_0^2 \lambda/3!$ we arrive at the Lagrangian $$ \mathcal{L}= \frac{1}{2} (\partial_\mu \phi) (\partial^\mu \phi) - \frac{m^2}{2} \phi^2 -\frac{\lambda}{4!} \phi^4 \tag{8} \label{8} $$ of the so-called $\phi^4$ model.
Choosing the physical mass $m_{\rm ph}$ of the associated scalar one-particle state $|p\rangle$ (with $p^2=m_{\rm ph}^2$) and the $\phi(p_1) \phi(p_2) \to \phi(p_3) \phi(p_4)$ scattering amplitude $\lambda_{\rm ph}:= |\mathcal{M}(s_0,t_0)|$ taken at some specific kinematical point characterized by the Mandelstam variables $s_0, t_0$ as our experimental input parameters, it is now our task to determine the corresponding theoretical expressions of these two observables.
Instead of taming the ultraviolet divergences by a "hard" momentum cut-off $\Lambda$ in the loop integrals, it is slightly more convenient to use dimensional regularization by formulating the theory in $d$ space-time dimensions, taking the limit $d\to 4$ at the end.
The squared physical mass $m_{\rm ph}^2$ can be extracted from the two point function \begin{align} \langle 0 |T\phi(x_1) \phi(x_2)|0\rangle &= \frac{1}{i}\int \! \frac{d^dk}{(2\pi)^d} \frac{e^{-ik(x_1-x_2)}}{m^2+\Sigma(k^2)-k^2-i \epsilon} \tag{9} \label{9} \end{align} as the pole of the momentum space propagator (see e.g. https://physics.stackexchange.com/q/835974 for further details) determined by the equation $$ m^2+\Sigma(m_{\rm ph}^2)- m_{\rm ph}^2=0. \tag{10} \label{10} $$ Expanding the self-energy function $\Sigma(k^2)$ around $k^2=m_{\rm ph}^2$, $$ \Sigma(k^2)= \Sigma(m_{\rm ph}^2) + \Sigma^\prime(m_{\rm ph}^2) (k^2- m_{\rm ph}^2)+ \ldots \tag{11} \label{11}$$ and defining the field renormalization constant by $$Z= \frac{1}{1-\Sigma^\prime(m_{\rm ph}^2)}, \tag{12} \label{12}$$ the momentum-space propagator takes the form $$ \frac{1}{i} \frac{1}{m^2+\Sigma(k^2)-k^2-i \epsilon} = \frac{1}{i} \frac{Z}{m_{\rm ph}^2-k^2} + \ldots \tag{13} \label{13} $$ in the vicinity of the pole.
Computing $\Sigma(k^2)$ up to one-loop order, one finds $$ \Sigma(k^2)=- \lambda \Delta(m^2) /2+ \mathcal{O}(\lambda^2), \tag{14} \label{14} $$ with the loop integral $$ \Delta(m^2)= \int \, \frac{d^d k}{(2\pi)^d} \frac{1}{m^2-k^2-i \epsilon}=\frac{i m^{d-2}}{(4\pi)^{d/2}} \Gamma(1-d/2) \tag{15} \label{15} $$ computed in dimensional regularization. Obviously, the Gamma function $\Gamma(1-d/2)$ exhibits a pole for $d \to4$ reflecting the ultraviolet divergence of the tadpole integral. Using eqs. \eqref{10}, \eqref{14} and \eqref{15}, one finds $$ m_{\rm ph}^2= m^2 +\frac{\lambda}{2} \frac{m^{d-2}}{(4\pi)^{d/2}} \Gamma(1-d/2)+ \mathcal{O}(\lambda^2). \tag{16} \label{16} $$ We observe that keeping $m$ fixed would lead to the nonsensical result of a divergent physical mass in the limit $d \to 4$. Instead, we have to take $m=m(d)$ such that $m_{\rm ph}$ remains finite in this limit. Inverting \eqref{16}, we find $$ m^2= m_{\rm ph}^2- \frac{\lambda}{2} \frac{m_{\rm ph}^{d-2}}{(4\pi)^{d/2}} \Gamma(1-d/2)+ \mathcal{O}(\lambda^2), \tag{17} \label{17} $$ where $(m_{\rm ph}^2-m^2) \lambda= \mathcal{O}(\lambda^2)$ was used.
The $S$-matrix element $$ \langle p_3, p_4\, {\rm out} | p_1, p_2 \,{\rm in}\rangle = (2\pi)^4 \delta^{(4)}(p_1+p_2-p_3-p_4) \mathcal{M}(p_1, p_2 \to p_3, p_4) \tag{18} \label{18}$$ of the process $\phi(p_1) \phi(p_2) \to \phi(p_3) \phi(p_4)$ is obtained from the (connected) four-point function by using the Lehman-Symanzik-Zimmerman (LSZ) reduction formula \begin{align} \langle p_3,p_4 \, {\rm out}| p_1,p_2 \, {\rm in} \rangle &=\left(\!\frac{i}{\sqrt{Z}}\!\right)^4 \int \! d^4x_1 \, d^4x_2 \, d^4x_3 \, d^4x_4 \, e^{-ip_1x_1} e^{-ip_2 x_2} e^{ip_3x_3} e^{ip_4 x_4}\\[5pt]& \; \; \;\times \left(\prod\limits_{i=1}^4 (\square_i+m_{\rm ph}^2)\right) \, \langle 0 |T\phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4)|0\rangle_c . \tag{19} \label{19}\end{align} To one-loop order, one has $Z=1+ \mathcal{O}(\lambda^2)$ from eqs. \eqref{12} and \eqref{14} and the expression for the invariant scattering amplitude is found as $$ \mathcal{M}(p_1, p_2 \to p_2, p_4)= - \lambda +\frac{\lambda^2}{2}\left(B(s, m_{\rm ph}^2)+B(t,m_{\rm ph}^2)+B(u,m_{\rm ph}^2) \right) \tag{20} \label{20} $$ with the three Mandelstam variables $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$ and $u=(p_1-p_4)^2$. In arriving at \eqref{20}, we have used $\lambda^2 B(s, m^2)= \lambda^2 B(s, m_{\rm ph}^2)+ \mathcal{O}(\lambda^3)$ valid at one-loop order. The function $B(p^2, m_{\rm ph}^2)$ is given by \begin{align} B(s,m_{\rm ph}^2)&= \frac{\Gamma(2-d/2)}{(4\pi)^{d/2}} \int\limits_0^1\! d \alpha \, \left[m_{\rm ph}^2-\alpha (1-\alpha) p^2\right]^{\frac{d-4}{2}} \\[5pt] &\to -\frac{1}{8 \pi^2 (d-4)}+ \; \text{finite terms}. \tag{21} \label{21} \end{align} Note that the divergent term in \eqref{21} is independent of $p^2$.
In contrast to the case of the mass, where the physical value is uniquely determined by the energy-momentum relation $p^2=m^2$, the definition of the physical coupling constant is convention dependent. Because of the relation $s+t+u=4 m_{\rm ph}^2$, the invariant amplitude in \eqref{20} depends in fact only on the two independent variables $s, t$. The observable differential cross section in the center-of-mass system is related to $\mathcal{M}(s,t)$ by the formula $$ \frac{d \sigma}{d\Omega}= \frac{1}{64 \pi^2 s} |\mathcal{M}(s,t)|^2 \tag{22} \label{22} $$ A possible definition of the physical coupling constant is thus provided by $$ \lambda_{\rm ph} := |\mathcal{M}(s_0,t_0)|, \tag{23} \label{23} $$ where $(s_0,t_0)$ is a conveniently chosen (but otherwise arbitrary) point in the physically allowed $s$-$t$ region. Using this definition, \eqref{20} leads to the relation $$ \lambda_{\rm ph}= \lambda \left\{ 1 - \frac{\lambda}{2} {\rm Re}\left[B(s_0,m_{\rm ph}^2)+B(t_0, m_{\rm ph}^2)+B(u_0,m_{\rm ph}^2)\right]+ \mathcal{O}(\lambda^2) \right\}, \tag{24} \label{24}$$ where $u_0=4 m_{\rm ph}^2-s_0-t_0$. The coefficient $\lambda=\lambda(d)$ must be chosen in such a way that the divergences in the loop functions (c.f. \eqref{21}) are compensated in the limit $d \to 4$. Inverting \eqref{24} finally leads to $$ \lambda=\lambda_{\rm ph} \left\{ 1 + \frac{\lambda_{\rm ph}}{2} {\rm Re} \left[B(s_0,m_{\rm ph}^2)+B(t_0, m_{\rm ph}^2)+B(u_0,m_{\rm ph}^2)\right]+\mathcal{O}(\lambda_{\rm ph}^2) \right\}. \tag{25} \label{25} $$
Edit: The following addendum addresses two questions raised in the comments below in some detail.
Using \eqref{25}, the scattering amplitude \eqref{20} can now be expressed in terms of $\lambda_{\rm ph}$, \begin{align} \mathcal{M}(s,t) &=-\lambda_{\rm ph}\bigg\{ 1-\frac{\lambda_{\rm ph}}{2}\bigg[\underbrace{B(s, m_{\rm ph}^2)- {\rm Re} \,B(s_0, m_{\rm ph}^2)}_{\text{finite}} + \underbrace{B(t,m_{\rm ph}^2)-B(t_0, m_{\rm ph}^2)}_{\text{finite}} \\[5pt] & \qquad \qquad \qquad \qquad+\underbrace{B(u,m_{\rm ph}^2)-B(u_0, m_{\rm ph}^2)}_{\text{finite}} \bigg] +\mathcal{O} \lambda_{\rm ph}^2) \bigg\}, \tag{26} \label{26} \end{align} showing that the scattering amplitude at all kinematical points $(s,t)$ is uniquely determined (to the given loop order) once it is known (measured) at one arbitrary reference point $(s_0,t_0)$ in the Dalitz plot (remember the definition of $\lambda_{\rm ph}$ in \eqref{23}).
The determination of the relations $$ m=m(m_{\rm ph}, \lambda_{\rm ph}, d), \quad \lambda= \lambda(m_{\rm ph}, \lambda_{\rm ph}, d) \tag{27} \label{27} $$ requires the computation of the two- and four-point function to the desired order in perturbation theory. This approach, although the most transparent one from a conceptual point of view, leads to rather cumbersome expressions (as we have seen, already at the on-loop level). On top of that, it depends also the convention used for the definition of the physical coupling constant $\lambda_{\rm ph}$.
In practical calculations, it is therefore more convenient to take advantage of a different approach, pioneered by Gerard 't Hooft in the early 1970s. It is based on the observation that in dimensional regularization, divergences appear as poles in the complex $d$-plane and the finite part of any divergent loop integral may be defined by removing these singularities by a certain prescription.
Taking $B(0, m^2)$ (defined in \eqref{21}) as an example, we have $$ B(0,m^2)= \frac{\Gamma(2-d/2) m^{d-4}}{(4\pi)^{d/2}}=-\frac{\Gamma(3-d/2)m^{d-4}}{(d-4) (4\pi)^{d/2}}, \tag{28} \label{28} $$ exhibiting a pole at $d=4$ with residue $-2/(4\pi)^2$. In the minimal subtraction scheme (MS) one defines therefore $$ B(0,m^2)_{r} \Big|_{\rm MS}:= B(0,m^2)+\frac{2 \mu^{d-4}}{(4\pi)^2} \frac{1}{d-4}, \tag{29} \label{29} $$ being finite for $d \to 4$. The renormalization scale or running (mass) scale $\mu$ in the second term is introduced in accordance with the dimension $[B(0,m^2)]=d-4$ of the first term. Expanding \eqref{29} around $d=4$, one obtains $$ B(0,m^2)_r \Big|_{\rm MS} =-\frac{2}{(4\pi)^2} \left[\ln \frac{m}{\mu}- \frac{1}{2} \Gamma^\prime(1) - \frac{1}{2} \ln(4\pi) \right]. \tag{30} \label{30} $$ The appearance of the ugly terms $\Gamma^\prime(1)$ and $\ln(4\pi)$ can be avoided in the modified minimal subtraction scheme ($\overline{\rm MS}$), where $$ B(0,m^2)_r \Big|_{\overline{\rm MS}}:= B(0,m^2) +\frac{2 \mu^{d-4}}{(4\pi)^2}\frac{\Gamma(3-d/2)}{(4\pi)^{\frac{d-4}{2}} (d-4)}= - \frac{2}{(4\pi)^2} \ln \frac{m}{\mu}+\mathcal{O}(d-4). \tag{31} \label{31}$$ Using $\overline{\rm MS}$, we can rewrite eq. \eqref{24} as \begin{align} \lambda_{\rm ph}&= \lambda \left(1+ \frac{3 \lambda \mu^{d-4}}{(4\pi)^2}\frac{\Gamma(3-d/2)}{(4\pi)^{\frac{d-4}{2}} (d-4)} \right) \\[5pt] & \quad -\frac{\lambda^2}{2} {\rm Re} \, \left[ B(s_0,m_{\rm ph}^2)_r +B(t_0, m_{\rm ph}^2)_r +B(u_0,m_{\rm ph}^2)_r\right] + \mathcal{O}(\lambda^3), \tag{32} \label{32} \end{align} suggesting the definition of the running coupling $$ \lambda_r(\mu):= \lambda \mu^{d-4} \left(1+\frac{3 \lambda \mu^{d-4}}{(4 \pi)^2} \frac{\Gamma(3-d/2)}{(4\pi)^{\frac{d-4}{2}}(d-4)}+ \mathcal{O}(\lambda^2) \right), \tag{33} \label{33} $$ being independent of any specific reference point on the Dalitz plot. Note also that the definition \eqref{33} ensures that $\lambda_r(\mu)$ is dimensionless even for $d\ne 4$. Expressing the running coupling in terms of $\lambda_{\rm ph}$ and taking the limit $d \to 4$ yields \begin{align} \lambda_r(\mu) &= \lambda_{\rm ph} \bigg\{ 1+ \frac{3 \lambda_{\rm ph}}{(4\pi)^2} \ln \frac{\mu}{m_{\rm ph}} \\[5pt] & \qquad \qquad +\frac{\lambda_{\rm ph}}{2} {\rm Re} \left[ \overline{B}(s_0, m_{\rm ph}^2) +\overline{B}(t_0,m_{\rm ph}^2)+\overline{B}(u_0, m_{\rm ph}^2) \right] + \mathcal{O}\lambda_{\rm ph}^2) \bigg\}, \tag{34} \label{34} \end{align} where $\overline{B}(p^2,m^2):=B(p^2,m^2)-B(0,m^2)$. Defining a mass scale $\mu_0$ by $\lambda_r(\mu_0)= \lambda_{\rm ph}$, the relation \eqref{34} between the running coupling $\lambda_r(\mu)$ in the $\overline{\rm MS}$-scheme and the physical coupling $\lambda_{\rm ph}$ defined in \eqref{23} can thus be written in the compact form $$ \lambda_r(\mu)= \lambda_{\rm ph} \left(1 + \frac{3 \lambda_{\rm ph}}{(4\pi)^2}\ln \frac{\mu}{\mu_0} + \mathcal{O}(\lambda_{\rm ph}^2) \right). \tag{35} \label{35} $$
Note that further aspects of this important topic can be found in https://physics.stackexchange.com/q/693546 and https://physics.stackexchange/q/408762.
