Motivation for introducing algebraic topology?

The best elementary justification for algebraic topology is that it provides a way of rigorously proving when some spaces are not homeomorphic. It's intuitively obvious, for instance, that the 2-dimensional sphere $S^2$ and the torus $T^2$ are very much different spaces. But to actually prove from scratch that every map $S^2\to T^2$ is either not bijective or has a discontinuous inverse is daunting. On the other hand, as soon as you know the first facts of homology theory or about the fundamental group you can write down a proof in just a couple of lines.

A somewhat more sophisticated motivation is that algebraic topology has the best tools for making sense of questions that are invariant up to homotopy-most simply, which spaces are homotopy equivalent. The fundamental group (and higher homotopy groups) and (co)homology theory (theories) are just as good at distinguishing non-homotopy equivalent spaces as non-homeomorphic ones, which at first sight is even harder to do. Now, you'll notice I haven't said anything about proving spaces are homeomorphic, or homotopy equivalent. The former is impossible in generality much beyond, say, 2-dimensional manifolds. But the latter is possible, at least in theory, for a kind of space called CW-complexes, which are thus a favorite of algebraic topologists.

The point, overall, is that algebraic topology provides one with discrete invariants that are more tangible material for writing rigorous proofs than the purely topological motivation for an idea-it's much harder to fully comprehend a space, especially in words, than a group associated to that space. This reflects the common pattern that algebra is more verbal and geometry more visual.

I can't comment very well on the dividing line between general topology and algebraic topology, because I don't know anything at all about modern topological research that's not either algebraic or geometric (i.e. about manifolds.)

In order to "understand the motivation behind introducing the algebraic machinery to topology" you need to go back to the history of the subject, and how it developed out of problems in complex analysis, as did general topology too. If you can get hold of it, I recommend "History of Topology", edited I M James, Noth Holland, 1999, particularly as a start the articles by S. Lefschetz on "The early development of algebraic topology" and by I M James on "From combinatorial topology to algebraic topology". I also recommend the Introduction to Lefschetz's "Introduction to topology" (but not the rest of the book, which many find confusing). You should not think that general topology arose first, but instead that they developed together.

A problem with the early work was to define precisely "boundaries" and "cycles", to obtain the rule "every boundary is a cycle"; this is the rule $\partial \partial =0$. It was Poincar\'e who developed the excellent trick of taking formal sums of oriented simplices in a simplicial complex. It was much later that Eilenberg introduced the ordered simplex and the rule we know and love $$ \partial = \sum_{i=1}^n (-1)^i \partial _i .$$ In fact the idea of taking "formal sums" derives from integration over various domains, i.e. of writing for convenience $$\int_{C} f + \int_D f = \int_{C +D} f .$$ The story is complicated and with many twists, and you should not think the story of conceptual development has ended. I do agree that the question is a good one, since the intuitions of earlier generations may not have been fully expressed in the current formulations. For example, the vision of the topologists of the beginning of the 19th century idea of higher dimensional nonabelian versions of the fundamental group was dismissed from the 1930s and on, essentially because of the so-called Eckmann-Hilton argument, showing "double groups" are just abelian groups. However it turns out that "double groupoids" are more complicated than groups, and this idea has quite a long way to run!

I mention that the word "groupoid" does not occur in James' book.

January 14, 2017 There is more discussion in this preprint.