The fire control group (a.k.a. the trigger group) is located inside the receiver of a firearm. The fire control group comprises those parts of the firearm that handle the motion of the trigger lever (springs, levers etc.), any trigger safety locks, the hammer and the sear (this is the part that holds the hammer back until the trigger is pulled firmly).
In the above image, we see the fire control group for a Ruger 10/22 rifle in a disassembled state with its various components.
While a person can buy the individual components from various sources, there are also several manufacturers (e.g. CMC Triggers, Geisselle, Timney, Tapco etc.) that make complete replacement trigger groups for various firearms, that are in a pre-assembled housing that just needs to be dropped in to the receiver.
A Timney Fire Control Group. Click on image to enlarge.
In the above image, we see a fire control group made by a manufacturer called Timney, designed to be used with AR-15 rifles. This part is designed for people who have very little gunsmithing experience and offers a better feel than the original AR-15 trigger. Timney also makes replacement fire control groups for other rifles, such as the Remington 700, Ruger 10/22. Mosin-Nagant M1891 etc.
Here's a video demonstrating the replacement of the entire fire control group with a new one. Note that as the user disassembles the old fire control group, he pulls out the parts individually (because the original parts were not assembled as a single component), but when he installs the new group, he installs it as a single component.
By the way, one of the major differences between a semi-automatic rifle and its fully automatic version (e.g. AR-15 and M-16) is the fire control group. This is what makes the AR-15 a semi-automatic rifle, whereas the M-16 is capable of selecting between multiple firing modes.