Very good question. There are a bunch of different uses, but basically it's used as a tool to measure how susceptible an ordinal is to Replacement. So, the basic idea is, consider an ordinal number β (or it can be a wellordered set, or whatever). Given a subset S⊆β, we say that S is cofinal in β when: for all x<β, there exists y∈S such that x≤y. Since β is wellordered, then S is also wellordered, hence S can be assigned an ordinal number order type...
The cofinality of β is the smallest order type of any cofinal subset.
So, suppose the cofinality of β is some other ordinal number α. This means that there exists a cofinal set S⊆β such that the order type of S is exactly α. Of course, since the entire notion of order type is based on bijections, there is also an equivalent notion of cofinality defined in terms of functions. Given ordinals α,β, and a function f:α→β, we say that f is cofinal in β when it holds that, for all y<β, there exists x<α such that y≤f(x)
Using this functional definition, we could equivalently define "the cofinality of β" as the smallest ordinal α such that there exists a cofinal function f:α→β. This definition in terms of functions is equivalent to the previous definition in terms of sets. A function is cofinal if and only if its image is cofinal.
The reason set theorists care about cofinality is basically for the same reason we care about cardinality. For example, if I tell you that a set S is (infinite and) countable, then you know that there is a bijection between S and the natural numbers. Because you have this bijection, you know a lot of other things about S. For example, the powerset of S is bijectible with the powerset of the naturals. Also, S can be partitioned into countably many finite sets (or finitely many countable sets).
If S is some abstract structure, then you also know there's a very good chance that S could probably be constructed using first-order Peano Arithmetic, and this knowledge might tell you something else interesting.
Cofinality is used for the same kind of purposes. Take for example the cardinal number ℵ[ω]. It's not hard to prove that the cofinality of this cardinal is exactly ω. This happens because the set {ℵ[n] : n<ω} is a cofinal subset of ℵ[ω], and it has order type ω. Because it has cofinality ω, then ℵ[ω] is accessible via countable Replacement. That is to say, there is a definable function with domain ω such that the limit of the function is ℵ[ω], and this function is simply the map n↦ℵ[n].
If you were in a context where you have access to countable Replacement, and also you have access to all the ℵ[n] cardinals, then you immediately know you also have access to ℵ[ω], simply because ℵ[ω] is the limit of a countable sequence.
The relationship between cofinality and cardinality is actually much tighter than this. In general, the cofinality of an ordinal is always a cardinal number, and more specifically it's always a regular ordinal. Regular ordinals are ordinals which are equal to their own cofinality, and thus they are immune to Replacement. It's also possible to prove that an infinite ordinal is a cardinal if and only if it's the limit of regular ordinals, so they are very closely related.
Cofinality also comes up a lot when dealing with large cardinal axioms, and it is precisely because of its relationship with Replacement. For example, we know that all successor cardinals are regular, and thus they are immune to Replacement from below. However, successor cardinals are mune to Powerset, since the cardinality of the powerset is always at least as large as the cardinality of the successor cardinal.
The only case in which a cardinal is simultaneously immune to Replacement and Powerset is when it's a strongly inaccessible cardinal. Equivalently, the inaccessible cardinals are precisely the regular limit cardinals....
Due to the ambiguity about the Generalized Continuum Hypothesis, there is a notion of weakly inaccessible (a regular cardinal which is a limit of Aleph numbers) and strongly inaccessible (a regular cardinal which is a limit of Beth numbers), but it's basically the same idea. The two objects are equiconsistent; weakly inaccessible cardinals become strongly inaccessible in the constructible universe, so it's just a matter of taste.
