Translation:the professor's computer
The ب in حاسوب is part of the root of the word (ح س ب like حسب "to calculate"). What makes this construction possessive is putting the person who possesses the computer second (the professor in this case), kind of like an adjective.
If you only wanted to say "her computer" though, all you need to do is is put ه at the end of حاسوب, like حاسوبه. "My computer" needs a ي, making it حاسوبي. Each possessive pronoun is a suffix like that.
"The professor's computer and my computer" حاسوب الأساذ وحاسوبي