A road bike groupset includes any parts involved in braking, changing gear or running the drivetrain. This includes the shifters, brake levers, front and rear brake calipers, front and rear derailleurs (sometimes called mechs), crankset, bottom bracket, chain and cassette.
Other components like pedals may also share the groupset's name although they're not considered part of the core groupset.
When you're buying a new bike, after the frame, the groupset is the second determining factor in value for money. Groupsets higher up the hierarchy will be lighter, may last longer, and in some cases may have improved functionality such as more immediate responses to shifting and braking.
Increasingly, the highest spec groupsets are electronic, where a battery and motors shift gears for you in response to electronic commands sent either via wires or wirelessly to the derailleurs.
There are three main manufacturers of groupsets and bike components. Shimano is the largest and best known, while the other two of the "big three" are Campagnolo and SRAM. All three manufacturers offer a range of groupsets at competing price points.