First, let's explain the Wii's system layout.
The Wii has two chips, an ARM chip and a PPC chip. All games, most homebrew (exceptions basically boil down to Mini and cIOSes), and the System Menu (which is
not an operating system) run on the PPC chip. Only one program runs at a time on the PPC, and the running program has full control of it.
The ARM chip runs the actual operating system of the Wii: normally IOS. IOS controls access to most of the Wii's hardware under most circumstances (applications using DVDX v2 can access hardware themselves). PPC applications have to talk to it to install channels, use the network functions, even to get Wii Remote data. Your Wii does not just have one IOS on it at a time, there are many different versions, such as IOS36, IOS60, and IOS9, though only one runs at a time.
Games eventually switch to using newer IOS versions. For example, a lot of third-party games used IOS21 for a period of time, but newer ones use IOS55 and IOS56. If you didn't have IOS55, (for example, because you still had an ancient System Menu) the System Menu would make you update before you play the game. If you tried to force the game to run on a different IOS, it would crash. This is not actually true in every case, but applications will not necessarily work if run on an IOS they are incompatible with. Most games will probably run on System Menu 3.4 through Gecko, but not all will.
To install an IOS, download Dop-IOS MOD v8 from the WiiBrew wiki. Launch that through the Homebrew Channel, and select IOS50 at the first screen. At the second screen, select IOSes.
DO NOT TRY TO INSTALL IOS51 ON YOUR WII UNLESS YOU ALSO INSTALL 60, 61, AND THE NEWEST SHOP CHANNEL. Also, installing IOS50 without updating the System Menu will brick your Wii, but if you pick IOS50 at the first screen, you won't be able to install IOS50. Install the IOSes you need, and you are good.