The chainset, also referred to as crankset, is the collective name for the front set of chainrings and crank arms, and nowadays the attached bottom bracket axle, which rotates round the bottom bracket shell on your bike.
Chainsets are manufactured with either one (single), two (double) or three (triple) chainrings and the more chainrings the more gears you have. If you want more chainrings up front to increase your spread of gears you will have to buy a complete new chainset as rings cannot be simply attached to your existing set up (you may also need to replace your front mech and gear lever as well so can become expensive).
The number of teeth on the chainring influences the amount of effort needed to turn the pedals so a small number of teeth requires little effort, easier to pedal, while a larger number of teeth increases the amount of effort needed, harder to pedal. Nearly all road bikes are sold with a double with either 39 x 53 teeth (standard) or 34 x 50 (compact) and good news is that these can be swapped easily without needing a new front mech or gear lever making it an easier way to make your gearing easier or harder depending on which chainset you currently have.
Crank arm length also varies from 165mm to 180mm in 5mm jumps so the length of your leg will determine which size crank you need. General rule is the taller you are the longer the crank arm you use and most bike manufacturers increase crank length with frame size for taller riders.
There are now many manufacturers of chainsets all with their own unique features so you will need to check compatibility with your bottom bracket and current gearing system before making any significant changes with your chainset. As a general rule, you can interchange chainsets from the same manufacturer without any issues as long as you stick to your existing amount of gears, i.e. 10 speed with 10 speed etc and the same bottom bracket system.