A first day cover is an envelope where the postage stamps have been cancelled on their first day of issue.
The first day of issue is the day on which a postage stamp or Postal/stamp card (ie PHQ) card or stamped envelope is officially put on sale, usually sometimes in a particular city, though sometimes in several cities simultaneously, nationwide or in a particular region.
There will usually be a first day of issue postmark frequently a pictorial cancellation, indicating the city and date where the item was first issued, and "first day of issue" is often used to refer to this postmark. Unofficial first day of issue postmarks can also occur when an individual stamp collector or dealer purchases the stamps in question from a post office in the first day of issue city and then takes them or send them (on that same day) to a post office in another city to have them cancelled.
Related to first day covers (FDC's) are event covers. Instead of marking the issuance of the stamp(s) they commemorate events both big (say, the launching of a new space shuttle) and small (the opening of a new regional attraction, for instance). Sometimes a design on the left side of the envelope (a "cachet") explains the event or anniversary being celebrated. Ideally the stamp or stamps affixed relate to the event. Cancels are obtained either from the location or from the Postal Service's local Cancellation Services Bureau or designated cancellation centre.