TITLE: Motivation for introducing algebraic topology?
QUESTION [17 upvotes]: What kind of topological questions does algebraic topology answer where point set topology is not enough?
Phrased differently:
Where is the line (or maybe intersection) between point set topology and algebraic topology? What is the distinction between the problems each deals with?
I want to understand the motivation behind introducing the algebraic machinery to topology.

REPLY [5 votes]: 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.