A very rare crystal-clear, straight-forward description with intuitive explanations:

"Now, whether or not it is knotted does not depend on how thick the rope is, how long the rope is, or how it is positioned in space. As long as we don't cut the rope, any kind of continuous deformation of the rope, such as moving it around, stretching it, bending it, and so on, does not change an unknotted closed loop into a knotted one. So, if we want to study the possible different ways a closed loop can be knotted, we want to ignore any differences related to all these various kinds of continuous deformations. When we ignore all those properties, what is left are called topological properties. So, while two closed loops of different sizes or shapes are geometrically distinct, they could be topologically identical. They are topologically distinct only if they can not be transformed into each other with any continuous deformation. So, in the context of knot theory, topology is the study of the properties of knottedness, which do not depend on the details of position, shape, size, and so on."

Answer by Tom McFarlane:

Rather than attempting to describe in layman's terms the meaning of Topology and Algebraic topology in all their glorious generality, I would illustrate these with a specific and very intuitive example: Knot theory.Suppose we have a closed loop of rope, i.e., a rope with its ends connected together. Such a closed loop could be a simple ring or it could be knotted up in various different ways:Now, whether or not it is knotted does not depend on how thick the rope is, how long the rope is, or how it is positioned in space. As long as we don't cut the rope, any kind of continuous deformation of the rope, such as moving it around, stretching it, bending it, and so on, does not change an unknotted closed loop into a knotted one. So, if we want to study the possible different ways a closed loop can be knotted, we want to ignore any differences related to all these various kinds of continuous deformations. When we ignore all those properties, what is left are calledtopologicalproperties. So, while two closed loops of different sizes or shapes are geometrically distinct, they could be topologically identical. They are topologically distinct only if they can not be transformed into each other with any continuous deformation. So, in the context of knot theory, topology is the study of the properties of knottedness, which do not depend on the details of position, shape, size, and so on.Now, algebraic topology is a way of studying topological properties by translating them into algebraic properties. In the case of knot theory, this might involve, for example, a map that assigns a unique integer to any given closed loop. Such a map can be very useful if we can show that it will always assign the same integer to two closed loops that can be continuously deformed into each other, i.e., topologically equivalent closed loops are always assigned the same number. (Such a map is called aknot invariant.) For example, if we are given two closed loops and they are mapped to different integers, then this instantly tells us that they are topologically distinct from each other. The converse is not necessarily true, since a map with poor "resolving power" might take many topologically distinct closed loops to the same integer. Algebraic topology in the context of knot theory is the study of these kinds of maps from topological objects such as closed loops to algebraic objects such as integers. These maps give, as it were, algebraic perspectives on the topological objects, and that is what algebraic topology in general is about.

What are topology and algebraic topology in layman's terms?