Topology is a branch of mathematics that studies the topological behavior of sets of points. I'll make an analogy with geometry. In geometry, we consider two triangles to be essentially the same, and to have the same geometric properties, once we know the lengths of the sides and the measures of the angles. We don't care about what colors they are, or what they are made of. these are not geometric properties. Two triangles are considered geometrically equivalent ("congruent") if they are the same size and shape, and if we can superimpose one on the other by moving it rigidly in space.
In topology, we don't care about size and shape, but about a certain more abstract and general property that is something like how the figure is connected to itself. Two objects are considered topologically equivalent if one can be bent, stretched, or squeezed until it looks like the other—but tearing is not allowed. All triangles are topologically equivalent, and all line segments are topologically equivalent, but triangles are not equivalent to line segments, since to turn a triangle into a line segment requires that you make a hole in it. The famous saying is that in topology a doughnut and a coffee cup are the same: you can put a dent into the doughnut, then enlarge the dent into the vessel of the cup, while the hole in the doughnut shrinks to become the hole in the handle of the cup. But there is no way to turn a sphere into a cup, because you would have to punch a hole in the sphere to make the handle.
Algebraic topology is a branch of mathematics which studies topological structures by mapping them to algebraic structures. Algebraic topology deals with certain special functions, called "functors", which take a topological object, such as a circle or a sphere, and turn it into an algebraic structures, such as the set of integers under the operation of addition, or the number 0 under the operation of addition, in such a way that if two topological objects get turned into different algebraic objects you can be sure that the two topological objects were different to begin with. Usually the algebraic structures are easier to reason about than the topological ones were, which is why we study algebraic topology in the first place.
Abelian groups are examples of algebraic structures. A group is a set, such as the even integers, together with a binary operation on the set, such as addition. The operation is required to satisfy certain properties. For example, it must be associative, which means that $a+(b+c) = (a+b)+c$ for every elements $a,b,c$ of the set. If the operation is also commutative, which means that $a+b = b+a$ for all $a,b$, then the group is an abelian group.
The phrase "algebraic topology can never have a non self-contradictory set of abelian groups" is nonsense.