Armoured Fist Squad
Armoured Fist Squads is used to refer to mechanized infantry, an Armoured Fist squad goes into battle mounted in a Chimera and forms the bulk of fully mechanized regiments. Their equipment is often the same as other squads but the addition of an armored transport gives them superior speed and tactical flexibility compared to their footslogging counterparts. They also tend to have a higher attrition rate as they are most often used against targets for which their Chimera is vital to success. It is quite common for individual Armoured Fist squads to be attached to non-mechanized infantry regiments to provide additional fast-moving support.
Sometimes an Armoured Fist Squad can drive in vehicles other than a Chimera. Usually a Leman Russ is picked when the need of heavier firepower is needed, other times it is a Hellhound to clear swathes of infantry. While the Armoured Fist Squad forms the main one-two punch, the regular guys on foot provide supporting fire as well as picking off stray targets and potential anti-tank squads hiding behind a bush or a rock.
In previous codexes, it was deemed powerful enough to be considered a limited unit, and with good reason; it was one of, if not the the most efficient use of points for the Guard, giving you a fair-sized squad along with the ever-ubiquitous Chimera, ready to take and hold an objective or lead the charge directly into the enemy's teeth.
The Armageddon Steel Legion is the most prominent user of these AFS as they field tanks and transports like Lasguns. Seriously don't be surprise if they flood the entire map with a swam of tracked vehicles and armored divisions.
Armored Fist Squads tended to be constantly underestimated in early editions. Guardsmen as a whole aren't big threats, but with the right situation can quickly prove pretty dangerous, and giving them the combination of mobility, heavy weapons, and armor that a Chimera provides can actually be pretty intimidating. For points, the Chimera is hard to beat, and Guardsmen, while fragile, pack a lot of Dakka at a decent range. As such, a Guardsman Squad with a Plasma Gun or Grenade Launcher, and Missile Launcher or Lascannon is basically equipped to engage almost anything you could encounter on the tabletop. Better still, half the unit could fire from the Chimera without ever having to disembark, and the Chimera's hardy enough to take some punishment, unlike many other transports.
This little feature meant that the Armored Fist Squad was limited to one per Infantry Platoon.
Ways Around the Restriction
There actually was multiple ways around the restriction of Armored Fist Squads, back in third edition. The most obvious was Doctrines; by cutting out options your army wasn't keen on using anyway, you could easily grab the "Mechanized" doctrine, and every squad in your army would get its own Chimera, even if they couldn't ordinarily get one, which tended to result in humor when you had a squad of Ratlings roaming the table and sniping from the roof hatches. This was the most common way to make a mech-heavy Imperial Guard force.
A similar, if more expensive way of getting around it was the "Grenadiers" doctrine, which allowed you to field Stormtroopers as a troops choice. These Stormtroopers couldn't deep strike or infiltrate but you could still take a dedicated transport, so you'd wind up with an even stronger version of the Fist squad with better infantry rifles. The main downside is that this cost almost twice as much. You could alternately do this by taking the "Veterans" doctrine (which made Veteran squads a troop choice) and cramming that in the resulting Chimera. This is a bit cheaper, but still much more expensive than the standard Armored Fist.
A third way, which was even more points-inefficient (but many armies still did) was take an allied Inquisition detachment and include a Storm Trooper Squad with a dedicated transport Chimera. This was the least common, because essentially it meant that the Guard were taking an allied detachment for a unit it already had access to, and Inquisitorial Stormtroopers tended to use Rhinos rather than Chimeras because they were cheaper, and the last thing its home codex needed was more anti-infantry weapons.
The Restriction Ends
With the 5th Edition Codex, the Imperial Guard lost the mandatory restrictions for Armored Fist Squads, and damned near anything could (and did) take a Chimera. The entire army could now be mechanized with no problems. Unfortunately, this edition saw the Armored Fist Squad lose its sparkle considerably, as the new Valkyrie and access to the Taurox saw the Chimera gradually become a second-stringer.
It still remains popular, however, especially given the oldfags on /tg/, and still remains one of the most efficient uses of points around, so there's that.