Qiaochu links to a really nice article by Timothy Chow that says a lot about the mechanics of how to go from a filtered complex to its spectral sequence. Two questions that remain are, (1) why do filtered complexes show up so much, and (2) is there anything that you could do with a filtered complex other than compute its spectral sequence?
VA gives a very general motivation for why filtered complexes show up so much. Probably it is just more general than what I was going to say. To discuss things more concretely, (I am told that VA's interesting answer is not a generalization of this paragraph.) For question (1), here are two general ways that filtered complexes show up: (a) you might get a filtered complex if you are interested in a chain complex that comes from a filtered object. For instance it could be a stratified topological space. A CW complex is stratified, but it has a special structure so that its chain complex doesn't need a spectral sequence. (It is an acyclic resolution as VA says.) If you have a stratification that you wish was CW but isn't, then you just have a general filtered complex. (b) There are some cases where just a single map or similar gives you a filtered complex. These may all be special cases of what Grothendieck described. For example, suppose that you are interested in the de Rham cohomology of a manifold. (This is then a certain resolution of the sheaf of smooth constant functions on the manifold.) Suppose further that the manifold is fibered over another manifold, or maybe is just foliated. Then differential forms are filtered by how parallel they are to the foliation.
Concerning question (2), there is an interesting result for filtered complexes over a field coming from representation theory. (I learned this some years ago in a discussion with Michael Khovanov.) The theorem in this case is that there is no more information in a filtered complex than in its spectral sequence. For simplicity, let's look at filtrations of length $n$. Then a filtered vector space is a representation of the $A_n$ quiver with all arrows in the same direction. (Each term of the filtration is assigned to a vertex of the quiver, and the arrows correspond to the inclusions.) The quiver also has other representations, but filtered vector spaces are the projective modules. A filtered complex is thus a projective complex over the $A_n$ quiver algebra. It is just like homological algebra over any other ring; you usually look at projective complexes. Since the $A_n$ quiver has projective dimension 1, it is easy to identify the indecomposable chain complexes of filtered vector spaces. There are two types, with two terms and with one terms:
$$0 to k^{(i)} to k^{(j)} to 0 qquadqquad 0 to k^{(i)} to 0.$$
In this notation, $k$ is the ground field and $k^{(i)}$ is $k$ in degree $i$. The first type of complex is valid if $i ge j$. If you compute the spectral sequence of an indecomposable complex, you will see that detects the first type of term at the $(i-j)$th page, and kills it on the next page. The second type of complex is of course the surviving homology. You can also go backwards and reconstruct the filtered complex from its spectral sequence.
Of course it is simplistic to only discuss filtered complexes and spectral sequences over a field. Nonetheless, roughly speaking spectral sequences are no more than a framework for analyzing filtered complexes.
VA asked for more details about the indecomposable modules, which is a fair request because I was quite cryptic about the relation. In particular, I use non-standard indexing in my own thinking about this. Unfortunately, I'm not sure that I can convert to standard indexing without making a mistake, so I won't convert. But I did change one thing above: I fixed the filtration degrees so that they are correct.
Suppose that $C = (C_n)$ is a complex of filtered vector spaces. Say that the filtration is increasing and indexed by $mathbb{Z}_ge 0$. The complex has two degrees: The chain degree $n$ and the filtration degree $k$. Suppose that $partial$ is a differential with chain degree $-1$ and filtration degree $0$. (This is where the numbering begins to be non-standard, although it makes sense from the point of view of quiver representations.) Then the theorem is that over a field, the two kinds of indecomposable complexes are those listed above, where the term $k^{(i)}$ has chain degree $n$, and the term $k^{(j)}$ (if present) has chain degree $n-1$. To be precise about what $k^{(i)}$ means, it is a filtration of the field $k$ in which the degree $j$ subspace is $0$ when $j < i$ and $k$ when $j ge i$.
The page $E^0$ is the associated graded complex. The page $E^r$ has a differential of degree $(-1,-r)$. When $r = i-j$, the differential $partial^r$ of the first type of indecomposable complex connects the "tip" of $k^{(i)}$ to the "tip" of $k^{(j)}$ and kills them both on the next page. The other kind of indecomposable complex has a vanishing differential, so the induced differential on every page also vanishes.
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