When I first got the game and not knowing where to start, I just went out and started looking. As I did so, I came across a book "The Last King of the Ayleids" and for some reason I kept it. After a while I ended up in Winterhold at the College and was given the quest to find this book. Needless to say I can't complete the quest even though I have the book. Can the quest be restarted? Or can any quest for that matter?

Nope.

If you're on the PC you can use the console command setstage . See http://www.uesp.net/wiki/Skyrim:Console for more details.
However I'm not sure if "restarting" the quest will solve your problem, since you already are at the start of your quest. Have you tried dropping the book and picking it up again? Or maybe following directions to find the book, then go to the area where you found it and drop and pick it up again? Hopefully that will do something.

WhiteWillow. I was thinking of dropping and trying to restart but I guess in the back of my mind I figured it wouldn't work but I will try it. Also, I knew there were console commands but had no idea as to where to find them. Thank you for the link. So here I go, we'll see what happens.

When I get items before given the quest I just exit the conversation with the quest giver and initiate dialogue again, this SHOULD bring up the new dialogue option with the "you mean this (enter appropriate quest item here)?" Are you saying this did not work?