Can I install AffiliateWP on a separate website?
You can if you are using a WordPress multisite installation. For more details see this article: Will AffiliateWP work on a multisite installation?.
Otherwise, know that AffiliateWP will only track referrals and visits on the site that it is installed on. This will be the same WordPress site where you have your e-commerce, membership or other supported integration. You can also have multiple integrations on the same site. The full list of supported integrations can be seen here: http://docs.affiliatewp.com/category/75-integrations.