A homeomorphism is generally considered to be an additive and continuous function. Additive refers to the following property:
g(A + B) = g(A) + g(B)
Also, two objects are considered homeomorphic if there is a homeomorphism between them.
A physical analogy to the homeomorphism is cutting a string, tying it any way desired, and gluing the ends back together. So, a homemorphism can be made of the unknot, which cuts it into a two-ended strand that can be tied and glued, leaving behind a knotted string. A special case of a homeomorphism is an isotopy, which is a homeomorphism that has an inverse. The physical analogy differs in that the string is not allowed to be cut or cross itself.
An even deeper look into the homeomorphism would be to write a function f(P, t), where P is a set of points and t, that returns another set of points. So, one can imagine setting P as the points on the unit circle and looking at a three dimensional representation of the output. Since time is a factor, the output would be animated. Also, because a homeomorphism is continuous, the animation would be "smooth" and no part or whole of the circle can "jump". The circle can stretch, shrink, twist, turn, and pass through itself. Gradually, the result can be a knot, but the result will always appear as a continuous curve. Now, the difference with an isotopy is that there is an inverse, so every point along the curve in every point in time of the animation corresponds to exactly one point on the original circle. Note that the general homeomorphism is allowed to cross itself, but that means that at some point in time two points of the original circle were moved to the same place. Therefore, there is no inverse function at that point. Thus, one can imagine a knot twisting, turning, bending, and stretching, but not allowed to cross itself in an isotopy. In such a case, a true knot cannot be turned back into the unit circle.