Division by zero represents one of several things in physics (not indended as an exhaustive list), in order of increasing problematicness:
1) You did your algebra wrong.
2) Your assumptions are wrong.
3) Your theory is wrong.
Examples of 1 abound. An example of #2 would be something like "What is the gravitational acceleration at the center of the Earth? Well, Gravitational force is Gm/r^2, and so at the center r=0 and it diverges". Obviously the incorrect assumption is that we can treat all the mass of the earth as concentrated at a point while inside its structure.
An example of #3 might be singularities which appear in Einstein's General Relativity. Many people believe these singularities are artifacts which merely appear due to the lack of a quantum theory of gravity, rather than physical objects. I suppose, in a way, this is a special case of #2 where the assumption is fundamental to your theory. But the essential difference is #2 is a modeling error, whereas #3 is a theoretical error.
Sometimes math analogies used to describe physics can have weaknesses in the way they are implemented. For example, if a direction on a 2d surface is described as a slope, such as y/x, then a line in the y direction has a slope of y/0. If the direction was described as a counter clockwise angle from the x axis (polar coordinate convention), then the problem doesn't exist.
Jeff Reid's response just reminded me of another thing that divide by zero could signify in a physical theory, which ranks more problematic than algebra but less than assumptions:
1.5) Bad choice of coordinates.
For example, schwarzschild coordinates are singular at the event horizon of a black hole, even though there is nothing particularly singular about this location. Of course, there are many other trivial examples you could make up with just applying some singular transformation to normal Cartesian coordinates, but I like this one :)