A greatcoat is a military style of coat similar to the trenchcoat; in fact, I'm not really sure what the difference is other than that a trenchcoat is worn in the trenches and a greatcoat is not*.
Usually made of heavy fabric, most are double-breasted, although some army ones are not. They are long, coming down to around the knees, and keep the wearer very warm.
I have two nearly-identical greatcoats, differing only in size, which I purchased on eBay (the first was the wrong size, and the seller never told me how to return it). They are of Royal Marines design, which means the fabric is navy and the buttons are Staybrite silver with the relevant insignia.
For a good view of lots of greatcoats, watch the Remembrance Sunday parade at the Cenotaph - all the military personnel present (which will include a large portion of the Royal Family) will be attired in greatcoats.
A greatcoat can provide a very dashing yet warm look, but you must be prepared for walking around in a very obvious piece of clothing, with a large number of shiny buttons on the front.
The RAF greatcoat has recently become very well-recognised as the trademark clothing of Captain Jack Harkness in the Doctor Who spinoff Torchwood.
*I have now been informed by rootbeer277
that a trenchcoat is specifically designed to be waterproof, whereas a greatcoat is designed for warmth.