The "recipient" of something is the person who receives it. In other words, it's the person who gets it. For example, the recipient of a package that's delivered through the mail might have to sign a form saying that they got the package.
Someone who gets an award is the "recipient" of the award. The word "recipient" is pretty formal, so it's OK for giving formal introductions like in the example above. A more casual way to talk about someone who's won an award is to call them a "___ winner":
She's a Pulitzer Prize winner.