What you expect to receive when you order this depends quite a lot on where you are (check Bun and Roll also).
Where I am at time of writing; I would expect either a toasted bread roll with sultanas, sliced in two horizontally or a very similar style roll without sultanas and suitable for adding savoury fillings or making a hamburger.
The term seems to be used interchangably here and would more commonly be called a bun or roll in Australia and apparently Scotland also.
As a savoury roll, it's a bread product often with seasoning on top such as sesame seeds. It's essentially what hamburgers come in except maybe somewhat flatter and wider than fast-food varieties and it comes in varieties such as multigrain, wholemeal etc in much the same ways as bread does in most Western countries.
Another form is a possibly sweetened version which contains sultanas (which the locals call currants though I would reserve that for dried sultanas). This version is probably the original and appears to be what is more commonly referred to in the rest of England.
Let them eat tea-cakes.
caveat: At this point in time only North Yorkshire has been surveyed, it is unknown if other parts of England use this term and North Yorkshire habits may not be representative of England in general.
This is a nodeshell no longer.
update:Oolong and BlueDragon say that in London, Bristol and Southern England this is only a current bun. wertperch expands that to all the (unspecified) parts of England he's visited. That style of bun is indeed a tea-cake in North Yorkshire (served toasted with butter) in addition to those listed above.
After the various messages I've received on this, I'm wishing e2 accepted photographs so I could prove it. One of the fish and chip shops in town has a basket of buttered, savoury bread rolls/buns and a sign saying 'Tea Cakes - 30p' (price may not actually be 30p). LPM suggests that it also refers to Sponge-like Cakes. I recall that it is also a name for Madeleine type cakes that are served with tea. Oolong further adds that something called "Tunnock's Tea Cakes" is "a delicious biscuit base topped with marshmallow and covered with real milk chocolate". Apparently they're something of a Scottish institution. His friend John claims they are the foundation of the British Foreign Office.