What is a spectral sequence?

I'd like to draw your attention to Timothy Chow's motivating short introduction to spectral sequences "You Could Have Invented Spectral Sequences".

A spectral sequence is a tool to get from the homology of the associated graded of a filtered chain complex to the associate graded of the homology of the filtered chain complex. The former will usually be known and the output is something that you wanted to know.

One reasonably general example: The homology spectral sequence of a bunch of inclusions of toplogical spaces - Given a filtering of a topological space $0=X_{-1}=X_0 \subset ....X_n \subset ... X$, one gets the singular filtered chain complex $S_*(X_0) \subset....S_*(X_n) \subset ...S_*(X)$. There is a notion of a spectral sequence associated to the chain complex $S_*(X)$ and the above filtration of the chain complex $S_*(X)$.

The associated graded of this complex is $Gr(S_*(X)):=\oplus_{p \geq 0} S_*(X_p,X_{p-1})$. There is an induced differential on the associated graded of this complex which allows one to make sense of $H_*(Gr(S_*(X))$. This group is called $E_1$ otherwise known as the first page of the spectral sequence associated to the filtered chain chain complex $S_*(X)$.

You should care about spectral sequences because this data will usually be known (the realization of what this is will vary from situation to situation). The 'answer' you are looking for, that you do not know, is $H_*(X)$.. When the coefficients of the singular chain complex are in a field, the output of the spectral sequence will be exactly $H_*(X)$. For more general coefficients, the output will be $Gr H_*(X)$ which by definition is the associated graded of the $H_*(X)$ with respect to the induced filtration of the abelian group $H_*(X)$ below:

$im (H_*(X_0) \to H_*(X) \subset im (H_*(X_1) \to H_*(X) \subset...im (H_*(X_p) \to H_*(X) \subset im (H_*(X_{p-1}) \to H_*(X) ... \subset im (H_*(X)\to H_*(X)$.

How useful a spectral sequence is will often hinge on how good a description there is of the differential graded abelian group $E^1$.

When $X$ is the total space of a fibration with fiber $F$ and basespace $B$, projection $p$, with filtering given by $X_0=F \subset .... p^{-1} B_i \subset...X$, with $B_i$ the $i$-skeleton of a CW structure for $B$, there is a description of the group $E_1$ with that depends only on the topological data of $B$ and $F$. Namely there is an isomorphism $E_1\cong C_*(B) \otimes H_*(F)$, where $C_*(B)$ is the cellular chain complex of $B$. Exercise(totally doable): Verify this for the trivial fibration $F \hookrightarrow F \times B \to B$ by using the definition the cellular chain complex - $C_*(B) := H(Gr(S_*(B)))$.

Serre's main insight was that this holds for general fibrations. Thus one can get information about homology of the total space from the cellular chain complex base and the homology of the fiber. For more information about this spectral sequence see the English translation of Serre's thesis "Homology of Singular Fiber Spaces" in Topological library part 3. If you email me I will scan you the 100 pages.

For the spectral sequence associated to a filtration of $0 \subset...S^*(X,X_p) \subset S^*(X,X_{p-1}) \subset...S^*(X)$ with the singular chain complex having coefficients in a field, see the third part of Bott and Tu's Differential Forms in Algebraic Topology.

Coming back to the general notion of a spectral sequence, (not just of a bunch of inclusions of topological spaces), we have examples where the $E_1$,does not have a good description in terms of known objects. In these situations the 'answer' will have a good description but one won't have a good way of finding it.

One example is where the 'answer' you are looking for is $[Y,X]$, the(sometimes) group of homotopy classes of maps from a topological space $Y$ into a topological space $X$. If you want the associated graded of this, you will need to use filtration of the group $[Y,X]$ giving rise to the group $E_2=H_{\text{first grading}} (Y , \pi_{\text{second grading} X})$. For more information about this spectral sequence see chapter 14 of Mosher and Tangora's Cohomology Operations an Applications to Homotopy Theory.

But one often doesn't know the information in the $E_2$ or $E_1$ group of this spectral sequence.

One can use a different filtration, called the Adams filtration, that is easy to describe in terms of known objects, but only yields the $2-$ primary part of $[Y,X]$. For more information about this, see chapter 18 of Mosher and Tangora's Cohomology Operations. This spectral sequence is called the Adams spectral sequence and is a powerful tool for calculating the stable homotopy groups of spheres.