Ce and ça both mean the same thing (this) in French, so what is the difference between them? When would someone use one over the other?
This answer will be about the pronouns, but note that ce (with variants cet, cette) is also a demonstrative adjective and as such will be followed by a noun and that ça should not be confused with çà which is an old synonym of ici.
Ça is an alternative form of cela; they can replace each other in most if not all contexts, but ça tend to be used more often (but not exclusively) when speaking, while cela tend to be used more often (but not exclusively) in writing. One case where cela is used more often both orally and in writing is when it is opposed with ceci.
Ce, ceci, cela and ça are the neutral demonstrative pronouns. They don't replace a noun (forms with celui, celle, ceux, celles are used in those cases) but something implied or (part of) a sentence.
When they are opposed, ceci is used for the nearer (in space, in time or in the discourse), cela (or sometimes ça) for the further. Alone, ceci is for something quite near and cela or ça for something quite far, but cela is also used for things near, especially for part of sentences which precedes the pronoun.
Usages of ce are more limited:
in ce que, ce qui, ce dont, ... then it stands for a thing (it used to be able to stand for a person, but that's rare nowadays) or a (part of) a sentence.
as subject of être, ce stands for what comes before (in that case it can be redundant and only used for emphasis), or the situation.
it is also sometimes used as the subject of pouvoir or devoir, when used as modal verbs (aka semi-auxiliaries).
there are some other usages, more or less literary (ce me semble), more or less fixed remnants of older structures (Ce disant, ...)
and it is sometimes replaced with ça for emphasis.