Firstbits refers to the practice of abbreviating a Bitcoin address such that only enough of the initial characters of the address are given so that the full address can be identified from the block chain. It also refers to the website http://www.Firstbits.com.
The term Firstbits was popularized by the Firstbits website, which was provided with the intent of a "Bitcoin address shortener", working similar to a URL shortener. Firstbits.com performs a lookup service, giving the full address from the block chain from the initial portion of an address. Anyone with access to a complete copy of the block chain can independently perform the same lookup done by Firstbits.com.
Any time a Bitcoin address is used for transactional purposes and appears in the block chain, its firstbits - or the prefix that uniquely identifies that address within the block chain - are considered assigned at that time. If another address is created and used with the same prefix at a later time, the firstbits of that new address will be at least one character longer, as the shorter prefix remains reserved for the address that used it first. No action on the part of any user is required for firstbits to be assigned - any transaction activity (receiving or sending) for an address for the first time results in firstbits becoming automatically assigned.
Unlike Bitcoin addresses, firstbits are deemed assigned in a case-insensitive manner.
Firstbits is generally considered to be a bad idea because it encourages transaction spam. This position is held by most, if not all, of Bitcoin developers.