Conversion cookie is more often a misnomer used for referring to the conversion tracking tag used to retrieve a post-click or post-impression cookie.
In this case, the conversion tag is placed on the page visitors see after they complete a conversion and the tag looks for an eventual cookie placed when the visitors have previously seen or clicked an ad. If a cookie is found the conversion is attributed to the ad.
Cookies and tag used for tracking conversion are managed by marketing vendors selling ads. These vendors -for instance Google AdWords or retargeters - make available to advertisers the single snippet of HTML and JavaScript code to be placed on the "thank you page" after a conversion for retrieving the cookie placed during the prior ad exposure or click. Conversions due to ads are then displayed for advertisers in the arketing vendor platform interfaces.