Here is another interesting class to throw in the mix in between CW-complexes and general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allowed to attach cells in any order. I think these have the homotopy type of CW-complexes, but are nicer point-set wise then the general class of things with these homotopy types. (See Tyler's answer).

They also show up in nature! If you have a manifold and a Morse function, then the handle body structure doesn't usually give you a CW-complex structure unless the critical values are ordered by index (i.e all index k critical points have the same critical value, say k). Otherwise you end up attaching cells in unusual orders.

However, a CW-complex has a natural filtration. This allows you to induct on the dimension of cells. It also allows you to construct certain easy spectral seqeuences. For example, the cellular homology complex comes from a sort of trivial application of this filtration. More generally the construction of the Atiyah–Hirzebruch spectral sequence uses this filtration. So it is nice to know something is a CW-complex, or at least is homotopy equivalent to it.

So to summarise:

1. CW complexes have nice point-set and homotopical properties.
2. CW complexes have nice computational properties (for example a useful filtration).
3. Knowing that $X$ is homotopy equivalent to a CW complex allows you to transfer computational (homotopy invariant) results about CW-complexes to $X$. For example: the Atiyah–Hirzebruch spectral sequence.

There are many other classes of spaces satisfying the first condition, but fewer also satisfying the second. But notice that once you have the class of CW complexes you can do all sorts of things for spaces that are only homotopy equivalent to CW complexes without them actually being CW-complexes. This is a subtle point. Once you have CW-complexes at your disposal you can prove results about things which are not themselves CW complexes, which would otherwise be difficult to prove.

Here is another interesting class to throw in the mix in between CW-complexes and general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allowed to attach cells in any order. I think these have the homotopy type of CW-complexes, but are nicer point-set wise then the general class of things with these homotopy types. (See Tyler's answer).

They also show up in nature! If you have a manifold and a Morse function, then the handle body structure doesn't usually give you a CW-complex structure unless the critical values are ordered by index (i.e all index k critical points have the same critical value, say k). Otherwise you end up attaching cells in unusual orders.

However, a CW-complex has a natural filtration. This allows you to induct on the dimension of cells. It also allows you to construct certain easy spectral seqeuences. For example, the cellular homology complex comes from a sort of trivial application of this filtration. More generally the construction of the Atiyah–Hirzebruch spectral sequence uses this filtration. So it is nice to know something is a CW-complex, or at least is homotopy equivalent to it.

So to summarise:

1. CW complexes have nice point-set and homotopical properties.
2. CW complexes have nice computational properties (for example a useful filtration).
3. Knowing that $X$ is homotopy equivalent to a CW complex allows you to transfer computational (homotopy invariant) results about CW-complexes to $X$. For example: the Atiyah–Hirzebruch spectral sequence.

There are many other classes of spaces satisfying the first condition, but fewer also satisfying the second. But notice that once you have the class of CW complexes you can do all sorts of things for spaces that are only homotopy equivalent to CW complexes without them actually being CW-complexes. Once you have CW-complexes at your disposal you can prove results about things which are not themselves CW complexes, which would otherwise be difficult to prove.

Here is another interesting class to throw in the mix in between CW-complexes and general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allowed to attach cells in any order. I think these have the homotopy type of CW-complexes, but are nicer point-set wise then the general class of things with these homotopy types. (See Tyler's answer).

They also show up in nature! If you have a manifold and a Morse function, then the handle body structure doesn't usually give you a CW-complex structure unless the critical values are ordered by index (i.e all index k critical points have the same critical value, say k). Otherwise you end up attaching cells in unusual orders.

However, a CW-complex has a natural filtration. This allows you to induct on the dimension of cells. It also allows you to construct certain easy spectral seqeuences. For example, the cellular homology complex comes from a sort of trivial application of this filtration. More generally the construction of the Atiyah–Hirzebruch spectral sequence uses this filtration. So it is nice to know something is a CW-complex, or at least is homotopy equivalent to it.

So to summarise:

1. CW complexes have nice point-set and homotopical properties.
2. CW complexes have nice computational properties.

There are many other classes of spaces satisfying the first condition, but fewer also satisfying the second.

Here is another interesting class to throw in the mix in between CW-complexes and general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allowed to attach cells in any order. I think these have the homotopy type of CW-complexes, but are nicer point-set wise then the general class of things with these homotopy types. (See Tyler's answer).

They also show up in nature! If you have a manifold and a Morse function, then the handle body structure doesn't usually give you a CW-complex structure unless the critical values are ordered by index (i.e all index k critical points have the same critical value, say k). Otherwise you end up attaching cells in unusual orders.

However, a CW-complex has a natural filtration. This allows you to induct on the dimension of cells. It also allows you to construct certain easy spectral seqeuences. For example, the cellular homology complex comes from a sort of trivial application of this filtration. More generally the construction of the Atiyah–Hirzebruch spectral sequence uses this filtration. So it is nice to know something is a CW-complex, or at least is homotopy equivalent to it.

So to summarise:

1. CW complexes have nice point-set and homotopical properties.
2. CW complexes have nice computational properties (for example a useful filtration).
3. Knowing that $X$ is homotopy equivalent to a CW complex allows you to transfer computational (homotopy invariant) results about CW-complexes to $X$. For example: the Atiyah–Hirzebruch spectral sequence.

There are many other classes of spaces satisfying the first condition, but fewer also satisfying the second. But notice that once you have the class of CW complexes you can do all sorts of things for spaces that are only homotopy equivalent to CW complexes without them actually being CW-complexes. This is a subtle point. Once you have CW-complexes at your disposal you can prove results about things which are not themselves CW complexes, which would otherwise be difficult to prove.