What kind of plastic are LEGO bricks made of?
Actually, it depends on the part.

Most parts are indeed ABS as it's a very sturdy plastic, but sometimes it's not suitable, or downright impossible to use. I've seen a presentation on quality issues given by a LEGO employee in LEGOworld a few years back, and as far as I remember there were between 10 and 20 sorts of plastic in use. The ones I remember are:


ABS (acrylonitrile butadiene styrene) for most regular bricks
PC (polycarbonate) for transparent bricks, as ABS can't be made transparent. Polycarbonate parts on the other hand have too much friction when in contact with other polycarbonate parts, making them extremely hard to disassemble. LEGO designers are forbidden to have some PC parts put together in models.
Rubber for tires and rubbery things - that's kinda obvious
A less sturdy plastic (can't remember the name) for parts which are more subject to being bent, for example studless technic beams.
As of  2018 plant elements will be made from a polythylene sourced from sugar cane.


And the list goes on. I know the presentation hadn't been made public, but some people took pictures, so if I ever find it back I'll add a link.

EDIT: still can't find it, but I know it was given at LEGOworld in 2006 by Bjarke Schønwandt who was Global Quality Manager at LEGO at the time (he may still be, I have no idea).