Weird, becuase when I copy your codes to Mathematica it still shows errors. So something must be wrong with my program. Maybe it's because I have Mathematica 8. I suppose you have newer version, don't you ?

In the future please post your code usind <> on the toolbar so we can copy it. It's a little more tricky:

s = NDSolve[{D[u[x, y], x, x] + D[u[x, y], y, y] - 3 u[x, y] == x y, 
   u[0, y] == 0, (D[u[x, y], x] /. x -> 2) == 0, u[x, 0] == 0, 
   u[x, 1] == 0}, u, {x, 0, 2}, {y, 0, 1}]

Plot3D[u[x, y] /. s[[1]], {x, 0, 2}, {y, 0, 1}]

In psudeo-math-english,

in the top you have del u/del x when x equals 2 is zero

while in the bottom you have u[2,y] is zero.

That was why I suggested you carefully compare the top and bottom half, character by character, to stumble onto this.

Exactly how to resolve that is another issue.

Since your teacher clearly told you to that you need to figure this out by yourself, I am very reluctant to tell you how to fix this.

I will give you one hint: Spend a minute concentrating on the idea that you know absolutely nothing about your problem, you have never seen this problem before, all you have is what you are about to see on the screen above this. When you have reached that state then compare the top half of your question with the lower grey screenshot of your question. Compare those a single character at a time. You are looking to find any tiny differences between the two.

If you find those then you need to figure out what the problem really is. Maybe it was just a mistake in your trying to desktop publish the top half of your question, if so then that probably solves nothing. Maybe it was a mistake in your trying to tell Mathematica what you wanted to solve. If that is the problem then you have an opportunity. That may involve figuring out how to tell Mathematica what you want AND in a way that Mathematica will understand you. I am guessing that won't be the way you think it should be written. If you think you know how to do that then try much simpler little experiments, things so simple that you know what the answer should be, but not so simple that you aren't even solving the same class of problem, to see if Mathematica understands what you are trying to tell it. But you need to figure that out.

Yes it is possible to solve this kind of problem using NDSolve. No I won't ruin your opportunity to learn from this. I urge you to not go ask somebody else somewhere else to just tell you the answer, you will lose everything if you do that. I hope it works out for you.