Imagine a square, a rhomboid and a circle:
And you ask: "Could I cut the rhomboid in such a way that I am able to put the cut parts together so that I get the square on the left?"
Of course you can! Just cut one of the the "corner pieces" and put it on the other side:
And now you ask: "Could I cut the square in such a way that I am able to put the cut parts together so that I get the circle on the right?"
And you think about it and realize that independently of how you cut the square into pieces, as long as you are making finitely many cuts, you won't be able to rearrange them into a perfect circle.
If you then think of the square, the rhomboid and the circle as sets of points, you could say that the square and the romboid would be equidecomposable, but the square and the circle not.
Creating an analogy for paradoxical sets is harder, but I'll give it a try:
Imagine you have a sphere made of a mystical, special material. You put the sphere into a grinder and grind it up into really fine powder. It now happens that this special material of which the sphere is made of allows you to let the powder which comes out of the grinder fall into two molds which have the exact same shape and size as the original sphere. To you surprise you observe that at the end of the grinding process, the two molds are filled up to the rim with powder, leaving you with two complete spheres, which is obviously paradoxical.
In relation to this analogies I want to try to explain this two terms with sets:
Two sets $A$ and $B$ are called equidecomposable if you can split up $A$ into a finite number of subsets, move this subsets around - but in a "rigid fashion", which means that you can not "deform" the subsets, like changing the relative positions of the elements of such a subset or bending the corner piece of the rhomboid so that it gets curved - and be left with the set $B$.
A paradoxical set is a set $A$ which you can split up into two subsets $A_1$ and $A_2$, "transform" each of the (infinite) elements of the subsets each in such a way that at the end of the transformation, each of the subsets are equal to your original set $A$ (this sounds crazy and total unintuitive, which is the reason why its called "paradoxical").