For a lens, the focal point is where rays refracted by the lens converge. The angle of refraction depends on the material of the two zones the rays travel through. In the ait that would be air and glass, and in the water they would be water and glass. Since the refractive indices of air and water are different (as we know by looking at a pencil in a gass of water), therefore the focal length of the lens will be different there.
There is a mathematical definition of focus too. Take a line, any line, and call it a directrix. Take also a point that is not on the line and call it a focus. Now consider the set of all points (the locus)whose distance from the focus point is k times the distance from the directrix line. Here k is any real number and we have
If k = 1 the locus is a parabola
If k < 1 the locus is an ellipse
If k > 1 the locus is a hyperbola.