Yes, you may use Amazon affiliate links to specific products, but not banners.
Advertising outside of WordAds at WordPress.com or infrequent Amazon product links is not allowed. For full details, please see:
http://support.wordpress.com/advertising/
http://wordpress.com/types-of-blogs/
If you require other advertising, you might be interested in a self-hosted (WordPress.org) installation. There are no restrictions on advertising if you setup a self-hosted WordPress.org blog instead of a WordPress.com blog, but there are some added responsibilities.
Please see this link to learn about the differences:
http://support.wordpress.com/com-vs-org/
We have hosting recommendations at the following link if you are interested:
http://get.wp.com/hosting/