On a graph what ways can it not be differentiable? I know at cusps it's not differentiable but are there other instances where it is not differentiable?

in general, it has to be continous at the point, and it has to have the same derivative approaching from the right, and from the left.

not continuous, but has the same derivative:
y= 3 for x<2
y= 5 for x>2
not continous, and does not have the same derivative.
y=-x for x<0
y=x for x>0

and a few folks allow functions that have multi values (shame on them), so that at a particular x, y can have more than one value, take a circle about the origin for example. It is better to not define these as functions, but I know some present these as functions, they are not differentiable unless one specifies exactly which part of it.