Assuming the given bounding box limits are in the same spatial reference system as the stored coordinates, and you know which spatial operator (intersects or contained by) you need:
SELECT * FROM my_table WHERE coordinates && -- intersects, gets more rows -- CHOOSE ONLY THE @ -- contained by, gets fewer rows -- ONE YOU NEED! ST_MakeEnvelope ( xmin, ymin, -- bounding xmax, ymax, -- box limits my_srid)
Alternatively, if you prefer the sound of "contains" (instead of "contained by") the WHERE clause should be flipped:
WHERE ST_MakeEnvelope (...) ~ -- contains, gets same fewer rows coordinates
PS: Given (by OP after the above was posted) that the records are simple points, I think that the difference between "intersects" and "containment" becomes very subtle, affecting only the points on the edges of the bounding box.