What type of character is Prospero? What is his personality like?
I suppose you could say that Prospero is what we would nowadays call a control freak. This is a man who likes being in charge. It's not surprising that he is this way, seeing as how he was unceremoniously overthrown as Duke of Milan. Stranded on a remote desert island, Prospero has painstakingly built his own little kingdom, where he gets to be in charge once more. Although the island doesn't belong to Prospero, he treats it as if it does. He lords it over the island's native inhabitants Ariel and Caliban, treating them like servants—and woe betide either of them if they should be foolish enough to step out of line.
At first, Prospero is a very bitter man, hell-bent on using his magic powers to wreak revenge on the men who betrayed him. Again, he's trying to regain some of the control he lost when he was overthrown and banished from his own principality. But though he is initially vengeful, Prospero does eventually come to realize that hate is a destructive emotion, which does him no good whatsoever. By the end of the play, he's grown in stature as a human being, developing the emotional maturity necessary to forgive his enemies, relinquish control of the island, and give up the practice of magic once and for all.
Prospero's character blends that of the nobleman, the father, and the scholar. He gives orders casually, and expects them to be followed. He is proud, and can hold on to a grudge for years (witness how long he plans his return to power). However, he loves his daughter deeply, and wants to protect her from all harm, and acknowledges Ariel's service at the play's end.