What are different ways in which I can use a two dimensional mathematical induction? I will also appreciate any examples of its use.
By this I mean the principle that will be used when I have to prove a theorem for all $(x,y)$ by showing that it is true for $(0,0)$ and also if (assumption) it's true for $(x,y)$ implies it is true for $(x+1,y)$ and also $(x,y+1)$.
Hope I understand your question correctly. You have to prove some statement S(n,m) for every n and every m, where n and m are positive integers. I usually do it like this:
1) I prove it's true for n=1 and every m.
2) I assume it's true for n=1,2,...,k (or for n=k only), and for every m.
3) I prove it's true for n=k+1 and for every m.
From this I conclude that it's true for every n and every m.
As in 1), 2), 3) I always do "every m", I would not call this two-dimensional though. Of course while proving 1) and 3), you're free to do another/nested induction on m. It could be that's what you meant in your question. Just make sure you don't use something which you haven't proved yet (it's easy to get confused and mess up by providing a flawed proof).