Memory affects video playback the same way it affects all computer performance. Yes, more memory might improve video playback quality (if it is stuttering), but there is no guarantee.
How much memory do you currently have, depending on the model, 3-4GB will be the max (and I'm guessing you already have at least 2GB). Not sure what performance increases you'll see.
If you want to know if you can upgrade your memory, do a google search for "macbook pro memory upgrade". I found this support article on Apple.com. http://support.apple.com/kb/ht1270.
As for what type of memory to purchase, that depends on what type of 15" Macbook Pro you have (15" Macbook Pro doesn't actually mean anything, they made dozens of different 15" models). If there's a part number or something you could maybe google-search on "macbook 123XYZ memory upgrade", or try contacting Apple support (though they'll probably direct you to overpriced Apple-branded memory).