First party cookies are cookies that are set by the site itself and can only be read by this site. They are commonly used for the functionality of the site (e.g. shopping cart). This doesn't imply that these cookies are allowed by EU law.
Third party cookies
Third party cookies are set by third parties which can be tracking or marketing companies (e.g. advertising networks). These cookies get re-read during a visit on other sites if they do business with these companies as well. Retargeting is good example of an application.
Third party requests
These are request that are made from a user to an external service. Despite the fact that these requests don't set any cookies, they can still transfer privacy information to third parties. Google Analytics for example works through third party requests.