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

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."