Originally called the Lord General's (Monck's) regiment of Foot Guards, they derive their popular title, which became their official designation in 1670, from the fact that the army with which Monck restored the monarchy crossed the Tweed into England at the village of Coldstream, and that his troops (which were afterwards, except the two units of horse and foot of which Monck himself was colonel, disbanded) were called the Coldstreamers.
The Coldstream Guards are part of the Household Division.
The uniform of the Coldstream Guards is unique in having its buttons arranged in pairs (in reference to the regiment being the second foot guards regiment). Captain Darling from the TV comedy Blackadder was a Coldstreamer; this can be recognised from his uniform.
Original, even self-contradictory information from the 1911 Encyclopaedia Britannica