Homeomorphic, homotopy equivalent and deformation retracts. How do I get a feeling for this? We have  homeomorphism, homotopy equivalences and deformation retracts ( which are a particular case of the latter). Now my problem is that I know what they all mean, but I have troubles to see them in real world objects. 
Imagining somebody would give me two objects and ask me whether they are related by one of the three maps,..... 
What am I able to do with an object in order to get the other one if they are homeomorphic to each other?
What else am I able to do with an object in order to get the other one if they are homotopy equivalent to each other? So what is the additional degree of freedom that this operation has more than pure homeomorphism?
Sure, there are some hints an object can give me in order to find out if they are / or may be homeomorphic / homotopy equivalent like the fundamental group, topological properties etc. , but I am here more interested in a geometric understanding of these two operations.
Edit: Since the question I posted under the one answer I got so far is referring very much to the question itself, I should add it here:
So intuitively one could say that shrinking or fattening something that has 'no volume'(like a point) to something with non-zero volume (like a ball) is a classical difference between homeomorphisms and homotopy types, right?- Could anybody try to explain somewhat more what a continuous deformation is, that we can use for homeomorphisms? I mean there are a lot of things that you can do to an object stretching, folding, cutting, twisting etc., could you try to give a few hints which ones are allowed and which ones not?
If anything is unclear, plese let me know.
 A: Homeomorphism is the ultimate topological equivalence. Two homeomorphic spaces are identical, topologically. That is, you can continuously turn one to the other, and back again, so continuous functions seem them as "identical".
Homotopy equivalence says roughly that two spaces have the same "shape". For example, $\mathbb{R^n}$ is homotopy equivalent for all $n$, because a line can be fattened to a cube, to a tesseract, etc (and thinned down to a point).
Likewise, a circle, and a cylinder (minus top and bottom) and a thickened cylinder are all homotopy equivalent. 
So one thing we know is that homotopy does not respect dimension, and that fattening or thinning does not change homotopy type. But that is not the only thing homotopy does! There are homotopy equivalent spaces where one space is not just a fattened version of the other. For example, the 3 dimensional Whitehead manifold has the homotopy type of $\mathbb{R^3}$ (and hence of a single point), but it is not a thickened version of $\mathbb{R^2}$.
In general, one would say homotopy is a weaker invariant that groups more spaces together, and that makes them easier to study. It generally reduces uncountable problems to countable ones.
Deformation retractions are just a simple type of homotopy equivalence that is easy to visualise. We image a thicker space shrinking down continuously onto a smaller subspace, while that subspace sits still. 
A: My answer Identification of points versus line drawn between points to a stackexchange question does give a picture of two spaces of the same homotopy type and not homeomorphic. This should be  start for you. 
