As SimonG mentions above, it will be a ceramic cartridge and there are loads of different ones. Screwfix do some ceramic cartridges that fit most taps, so you'll 'probably' be able to replace it but may be a bit of a pain.
I'm sure that screw you undid is all that is holding the handle on. It must be seized, as SimonG suggests, use a block of wood and pry it off. Make sure the force is concentrated on where the actual sticky out handle part meets the cylindrical cover part or you may snap something.
Have you tried undoing the handle on the other side, it may be easier and will show you what you'd expect to see on the side that's giving you trouble. Cartridges are generally sold in pairs (they are at Screwfix) so do both sides at the same time, the other side is likely to be in a similar shape and may start dripping soon anyway.
Or buy another tap.
If you go for another cartridge take the old cartridge and handle with you. You can compare the size of the cartridges and be sure your handle will fit on the splines of the new one.
