Answer by Alon Amit:
Let's do Topology first, shall we?
Topology is (very roughly) the study of shapes that can be stretched, squished and otherwise tortured while keeping near points together. It is sometimes described as the study of deformations were no tearing is allowed, but this is somewhat misleading: you may tear to your heart's content as long as you patch things back together later. The Dehn Twist is an example of a legal topological move that cannot be achieved without cutting and pasting.
If our proverbial layperson is familiar with plane geometry, we can put it this way: the central object of study in plane geometry is congruence. Two shapes are congruent when one can be mapped to the other via a rigid motion: sliding it along, rotating it, or reflecting it. No deformations, expansions, or other twists are allowed. So in geometry we can talk about angles, for example, since angles don't change when you slide and rotate. Congruent triangles are ones that are the same except for a possible translation (sliding), rotation and maybe reflection (taking a mirror image).
Topology is the same thing, except that it's different. It's different in two ways: First, we're not limited to simple polygonal shapes in the plane – we're dealing with any "shape" (topological space) for which we have a notion of "nearness". Intuitively it's fine to think of familiar shapes in the plane, space or (if your imagination is up to it) 4 dimensions and beyond. In fact, topology is a lot more permissive than this, and allows for spaces that look like nothing that fits into any number of dimensions. Here's a rather tame example – trust me, it gets a lot wilder than this.
Or even this.
You can't talk about topology without showing a picture of a torus. Here's a torus. It's just one of the shapes topologists are fond of.
The second way topology differs from plane geometry is that the notion of "being the same thing", instead of congruence, is called topological invariance, which is formally defined as a bi-directional mapping of one shape to another which preserves this notion of points being near each other. It's ok to stretch and twist, it's ok to tear and paste back, but it's not ok to make a hole and keep it there, or to otherwise break things apart.
So, topologically, this, too, is a torus:
Topology is the study of those shapes and the ways we can classify them, check when they are and aren't the same, and – no less important – how this notion interacts with other parts of mathematics such as complex analysis, functional analysis, geometry and even number theory and combinatorics.
If you have a rubber band, you can use it to form a circle, an ellipse, or a square. Those are very different shapes geometrically, but from the topological perspective they are the same.
So what is algebraic topology? Well, in many ways the permissive nature of topology makes it hard to tell things apart. It's not entirely easy to show that the plane and 3d space are different topologically, which sounds a bit silly but it's a fact. Showing that 3d space isn't 4d space is even harder.
One of the ways people found to deal with those difficulties is to create gadgets (officially called functors) that map topological spaces into objects that are easier to handle – algebraic objects like vector spaces and groups. If our layperson doesn't know what those are, we can get some headway by just declaring that our gadgets map topological spaces into numbers, or lists of numbers.
We then show that if two topological spaces are the same (topologically), they will get the same numbers attached to them. Contrariwise, if two spaces are tagged with different numbers, they cannot be the same.
Defining those gadgets isn't entirely trivial. Classically this was done via things called "homology" and "homotopy" groups. The latter ones can be roughly described as counting the number of distinct ways you can put a closed rubber band inside your space. Two such rubber bands aren't considered distinct if one can be deformed into the other inside your space.
Here's a rather bad picture of a rubber band on the surface of a torus.
Thus, for example, take the plane minus a point, and take 3d space minus a point. In the latter, all rubber band placements are the same – there's complete freedom in moving a rubber band about in 3d space even when a point is missing. But in the plane, a rubber band that wraps around the missing point cannot be deformed into one that doesn't. Hence, plane-minus-point is not space-minus-point, from which it follows that plane isn't space.
 There aren't good animations of a Dehn Twist on YouTube (the ones that exist are more confusing than helpful). Take a donut, cut it open as you would if you wanted to make a cylinder, twist one end 360 degrees as if you're wringing it, and paste it back together so it looks like the original donut – but most points on its surface aren't where they were before.
 In fact, angles are much more resilient than this – they don't change even if you allow much more flexible transformations than rigid ones. Such transformations, which preserve angles, are called conformal. But in topology we don't really care about angles.