It's interesting to consider this problem from a complex analysis point of view. Of course, Nyquist's stability criterion as used in fibonatic's answer relies on results from complex analysis.
Let's first define the winding number. Let $f(z)$ be a complex function that is analytic in the interior of a closed contour $C$ except for a finite number of poles (i.e., $f(z)$ is meromorphic). We assume that $f(z)$ is analytic and non-zero on $C$. The winding number (with respect to the origin) is the number of times $w=f(z)$ winds around the origin of the $w$-plane as $z$ moves along the contour $C$ once. The winding number is given by the integer value
$$\frac{1}{2\pi}\Delta_C\arg f(z)\tag{1}$$
where $\Delta_C\arg f(z)$ denotes the difference between the final and the initial arguments of $f(z)$ as $z$ describes the contour $C$ one time.
The argument principle states that the winding number equals the number of zeros $Z$ minus the number of poles $P$ of $f(z)$ inside the contour $C$ (counting multiplicities):
$$\frac{1}{2\pi}\Delta_C\arg f(z)=Z-P\tag{2}$$
I adopt the more common notation $1\color{red}+H(z)$ for the denominator of the closed-loop transfer function. Since we're interested in stability, we choose the unit circle $|z|=1$ as our contour $C$.
If we assume that $H(z)$ is (strictly) causal and stable, it must have all its $N$ poles inside the unit circle, where $N$ is the filter order. From the argument principle $(2)$ we get
$$\frac{1}{2\pi}\Delta_C\arg \big\{1+H(z)\big\}=Z-P=Z-N\tag{4}$$
where $Z$ and $P$ are the numbers of zeros and poles inside the unit circle, respectively. Since the total number of zeros equals the filter order $N$, the absolute value of $(4)$ equals the number of zeros outside the unit circle.
For stability of the closed-loop transfer functions we require all zeros of $1+H(z)$ to be inside the unit circle. Hence, we require
$$\frac{1}{2\pi}\Delta_C\arg \big\{1+H(z)\big\}=0\tag{5}$$
as $z$ moves along the unit circle. The requirement that the argument of $1+H(z)$ returns to its original value after $z$ has moved around the unit circle once is equivalent to requiring that the trace of $1+H(z)$ in the complex plane does not encircle the origin, because each encirclement of the origin would mean an increase (or decrease) of the argument by $2\pi$.
The trace of $1+H(z)$ will not encircle the origin if $|H(z)|<1$ for $|z|=1$. This is a sufficient condition which we could also have obtained by applying Rouché's theorem.
A less restrictive, but still sufficient condition ensuring that $1+H(z)$ does not encircle the origin is $\textrm{Re}\big\{1+H(z)\big\}>0$ for $|z|=1$, i.e.,
$$\textrm{Re}\big\{H(e^{j\omega})\big\}>-1,\quad\omega\in[0,2\pi)\tag{6}$$
Note that $(6)$ is not a necessary but a sufficient condition. A necessary and sufficient condition would be
$$H(e^{j\omega})>-1,\qquad \textrm{for }\omega\in[0,2\pi)\textrm{ such that }H(e^{j\omega})\in\mathbb{R}\tag{7}$$
because requiring the trace of $1+H(z)$ to only cross the positive real axis but never the negative real axis guarantees that there are no encirclements of the origin.
Examples:
I'll give examples of FIR transfer functions $H(z)$ all of which lead to stable closed-loop systems. All transfer functions satisfy at least one of the three stability criteria discussed above:
- $\big|H(e^{j\omega})\big|<1$
- $\textrm{Re}\big\{H(e^{j\omega})\big\}>-1$
- $H(e^{j\omega})>-1$ for all $\omega$ such that $H(e^{j\omega})\in\mathbb{R}$
Criterion 1 is the strictest, and criterion 3 is the least strict. Criteria 1 and 2 are sufficient but not necessary whereas criterion 3 is necessary and sufficient.
The first filter is given by
$$H_1(z)=0.4z^{-1}-0.4z^{-2}-0.3z^{-3}+0.2z^{-4}$$
It satisfies $|H_1(e^{j\omega})|<1$, and, consequently, it also satisfies the other two criteria. All zeros of $1+H_1(z)$ are inside or on a circle with radius $0.84$.
The second filter is
$$H_2(z)=0.5z^{-1}-0.4z^{-2}-0.5z^{-3}+0.2z^{-4}$$
It violates criterion 1 but it satisfies criterion 2 (and 3). Its maximum magnitude on the unit circle is $1.21$, and all zeros of $1+H_2(z)$ are inside or on a circle with radius $0.92$.
The last filter is
$$H_3(z)=0.5z^{-1}-0.4z^{-2}-0.6z^{-3}+0.3z^{-4}$$
It violates criteria 1 and 2 but it satisfies criterion 3. Its maximum magnitude on the unit circle is $1.37$, and its minimum real part on the unit circle is $-1.08$. All zeros of $1+H_3(z)$ are inside or on a circle with radius $0.99$.
The figure below shows the complex traces of the three corresponding denominator polynomials $1+H_k(z)$, $k=1,2,3$. 
Clearly, none of them encircles the origin, as required for stability. It can be seen that the imposed stability conditions become less strict from left to right. The real part of $1+H_3(z)$ even becomes negative, but the trace crosses the real line at positive values, as required by condition 3.
