At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other areas of mathematics, especially, in geometry(I would say almost in all geometry).
I'm not an algebraic topologist myself, so I know only basic techniques. However, I'm intrigued by modern tool in homotopy theory. For example, we have simplicial homotopy theory, where one studies simplicial sets instead of topological spaces.
As far as I understand, simplicial techniques are indispensible in modern topology. Then we have axiomatic model-theoretic homotopy theory, stable homotopy theory, chromatic homotopy theory. Recently, we got a topological version of algebraic geometry, namely spectral algebraic geometry which is proved useful in studying topological modular forms.
But one may wonder what is it for? Those are really fancy and sometimes beautiful tools, but what are exactly the questions modern algebraic topology seeks to answer? Because It feels it's really not part of topology anymore, it's more as topology now is a small part of algebraic topology/homotopy theory.
So, I would like to hear about goals and perspectives of modern homotopy theory from those working on it. I hope this question might be useful to someone else.
Best Answer
While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still an especially important place in our heart. Especially when we say algebraic topology instead of homotopy theory. This says that while all of category theory and all of homological algebra belongs to the study of $(\infty, 1)$-categories, this is not where our aim is.
The roots of our subject lie in the study of nice spaces like manifolds. Important questions are:
The coarsest useful equivalence relation for the classification of manifolds is bordism and this is also the basis of most other classification results (those using surgery theory). Computing bordism groups was an important topic in earlier algebraic topology and was done succesfully for some flavors rather early ($\Omega_O$, $\Omega_U$, $\Omega_{SO}$,...). But one of the most important variants, both theoretically and from the viewpoint of clasification of manifolds, is framed bordism. By an old theorem by Pontryagin, the framed bordism groups are isomorphic to the stable homotopy groups of spheres, connecting it to the second question.
One can say that much of algebraic topology was invented or can be used to study the stable homotopy groups of spheres. One of the most recent spectacular advances in algebraic topology was the solution of (most of) the Kervaire invariant 1 problem by Hill, Hopkins and Ravenel about framed manifolds/stable homotopy groups of spheres. They used a tremendous amount of stuff to solve this classical problem: equivariant topology, chromatic homotopy theory, spectral sequences, orthogonal spectra, abstract homotopy theory, ...
Likewise topological modular forms $tmf$ have important applications to the stable homotopy groups of spheres and also to string bordism. And to really understand $tmf$, you have to study some spectral algebraic geometry.
I do not want to say that all of algebraic topology still directly aims at classical questions. As soon as we see an interesting structure, we also study it for its own sake; new phenomena need explanations and developing abstract frameworks is also fun. But like in the relationship between mathematics and physics, sharpening our tools and exploring by pure curiosity can be quite useful for the classical questions. When people replaced older, in some aspects more clumsy models of spectra by symmetric and orthogonal spectra, they probably didn't have in mind any direct applications to framed manifolds. But what Hill, Hopkins and Ravenel did would have been much harder without these tools in their hands.