The Action Replay Mk VI cartridge is a printed circuit board inside a small, red case with two red buttons sticking out of the back. It plugs in the cartridge port on the back of a Commodore 64 computer. The left button freezes the C64 and brings up a menu, whereas the right button invokes another menu that you see when the computer's first booted.
It has many, many uses (mainly utilities for everyone and cheats for gamers), but I'll stick to the ones that are probably most useful to an average C64 user here:
Sprite Killer
The way most C64 games work is that the sprite you play will die when it touches a certain other sprite (such as a bullet or an enemy) or a certain background character (say, some spikes or acid). You can use this to stop the game noticing when such collisions between a player sprite and other sprites or background characters occur. Hence, it makes you invincible.
Pokes
Pick up pretty much any old C64 magazine and you'll probably find a cheats page stuffed with pokes. This will be in the form of POKE 53280,1. The first number is the location in memory of a specific byte, and the second is a number between 0 and 255, which that byte will be changed to. This can often be used to gain infinite lives. If you have such a poke, here's the place to enter it.
Pokefinder
If you don't have a poke as mentioned above, this is the option for you. Simply tell the cartridge how many lives you have, resume the game and lose a life. When you next freeze the game, it will tell you what the correct poke is to gain infinite lives.
Backup
This is arguably the best feature of the cartridge, and doesn't just apply to games. The entire memory of the C64 is compressed and saved to either disk or tape. This has two main advantages over loading software the way you're supposed to. The first is that it saves the exact state the machine is in, be it the last level of a game or a certain part of a multipart demo (the latter being useful if you want to save a disk-based demo to tape, where the loading of each separate part won't work properly as the demo automatically tries to load from the wrong device). The second advantage is efficiency. Often, the backup of the computer's state will take up considerably less space on the tape or disk, and therefore will also load much more quickly, than the original software loading the way it was designed to.
Directory
This is a simple directory listing of a disk. The standard method is much more complex: LOAD "$",8 to load the directory listing as if it was a BASIC programme, then LIST to actually view it (what were Commodore thinking?).
Disk Copy
Pretty self explanatory. I can only imagine how difficult this would be without the cartridge. You can copy from and to the same device if you're willing to switch between the disks an awful lot - they have more space than the C64 has RAM.
So basically, this cartridge is extremely useful to pretty much any C64 user who wants to spend less time waiting for things to load, performing directory listings and other simple tasks, and especially if they want to cheat at games.