In the stamp collecting community, a cover is usually referred to as an envelope, usually with a stamp on it, that has traveled through the mail. There are many variables of this basic definition.

The scan above is a good example of a fairly standard collector Naval Cover. It has traveled through the mail as you can see from the little edge bends and minor folds in the upper right corner. It has a postmark (often referred to as a "cancel") that notes the name of the ship and a date and year. In this case, a city is listed in the cancel as well. The design on the left of the envelope is known as the "Cachet", an area of artwork and/or print usually related to the cancel, ship or the date - in this case, the red text commemorates a special date - the settlement of Savannah, Georgia. In the lower right is the address. Not all Naval Covers have cachets, cities listed, or even the name of the ship in the cancel.

The basic elements of this cover: the postmark (with city, date and year) the cachet and the address, usually apply to other areas of cover collecting including Event, First Flight, Highway Post Office, Advertising Covers and others.

The second scan above is an example of an unaddressed cover. It probably did not go through the mail (as there is no address or remnant of an address label and the condition of the cover is pristine) but it is considered a cover as well.
This trend (of unaddressed covers) has become popular during the last 50 years or so in the areas of First Day Cover and Pictorial Cancel Collecting.