Think of what an electric motor is. You run an electric current through a coil of wire which in turn surrounds a magnet. A force is generated which can be harnessed to turn a fan blade, or a vacuum cleaner brush, whatever.
It works in the other direction too.
If you take a magnet and move it in and out of a coil of wire, an electrical current will be generated in the wire. Kind of like a reverse electric motor. That's what a phono cartridge is.
The stylus rides along a record groove on a little metal cantilever. It moves with the vibrations in the record goove, which should be an analog of the original music signal. The little vibrations cause a magnet and a coil of wire to move in relation to one another. That tiny electrical signal, which should also be an analog of the record groove vibrations, is then transmitted to the amplifier.
With an MM cartridge, it is the magnet that moves. The magnet is attached to the opposite end of the cantilever that the stylus is attached to. The coil of wire is fixed and is in the body of the cartridge.
With a MC cartridge, the coil of wire is attached to the end of the cantilever and it is the magnet which is fixed in the body of the cartridge.
So for both cartridges, you have the same thing: a magnet and a coil of wire. One is fixed and the other moves, thereby generating an electrical signal.
What are the advantages and disadvantages of each?
One advantage of a MM cartridge is that it is easy to replace the stylus. Since the magnet is not attached to anything in the body of the cartridge, you can pull the stylus/cantilever assembly out of the cartridge and replace it. You cannot do this with a MC stylus because it is the coil of wire which is attached to the cantilever, which is then attached to the body of the cartridge. If you pulled it out, it would break the wire. So to replace the stylus, it has to be sent back to the factory to be rebuilt or "retipped". Expensive.
Another advantage of the MM cartridge is that it is cheaper to manufacture. You don't have the added construction complexity of attaching a coil of wire from the cantilever to the cartridge body. Again, expensive.
Well, why bother with MC cartridges? As it turns out, they have advantages over MM cartridges on other points, not in cost but in performance.
We would like not to have a stylus/cantilever being weighed down by a magnet, which is a chunk of metal. The weight could inhibit the ability of the stylus/cantilever to move with record groove vibrations, some of which are in the order of billionths of an inch. Rather than put a realtively heavy magnet on the cantilever, if you put a lighter coil of wire on the cantilever,it will be much more nimble and able to respond to more subtle record groove vibrations.
There is another advantage to a MC pickup. You can use a smaller coil of wire than with a MM pickup.
If you have a small magnet on the MM cantilever in to avoid weight problems, you need a larger coil of wire in the cartridge to generate sufficient electrical signal.
With a MC cartridge, you can use a small coil of wire on the cantilever, which makes it light and nimble, and then put a relatively larger magnet in the cartridge body in order to generate sufficient electrical signal. It's better to have the relatively heavy magnet in the cartridge than on the cantilever, although you still don't want it too big and heavy in the cartridge either. But here's the other benefit of a small coil. It's becoming a bit technical, but here it is. A coil will cut off high frequencies. It's a filter. So a MM cartridge acts like it has a buit in high frequency filter. You can compensate for this, but it's causing more problems that have to be dealt with. On the other hand, if you have a small coil, as in a MC cartridge, the fitering effect is decreased. And you have the other advantage of adding less weight to the cantilever. But again, the finer the construction, the more expensive it is.
Because the MC cartridge has such a small signal, due to the coil being so small, the more likely it is to be overwhelmed by hiss, or by the noise floor of your other components. Although the MC is more sensitive to record virations, it follows that it is also more sensitive to other noise as well. So, although a MC is a significant upgrade in performance over a MM, the rest of the system must be commensurately good or you've wasted your money by reproducing hiss and noise as well as more subtle signal, which may in fact be overwhelmed the increased hiss and noise.
Because the MC produces a smaller signal than a MM, it needs extra amplification. So phono preamps have a separate setting for MM and MC preamps to reflect the extra boost needed fo the MC. If you use a MM phono section for a MC cartridge, it will likely only produce a whisper in volume.
One last point. There are high output and low output MC cartridges. A high output MC cartridge is an attempt to have the advantages of both MM and MC cartridges. It is a MC cartridge so it is more nimble than a MM cartridge. However, it uses a larger coil of wire than a low output MC cartridge. With a larger coil, it may produce enough output than it will work with a MM phono section. The downside is that you start adding weight back on to the cantilever, and the filter effect will increase as the coil gets larger. In other words, you're reintroducing the reasons you went from MM to MC in the first place. Still, it's a a trade-off you may prefer.
Like Elizabeth says, if you're a neophyte, use MM. There are very good MM cartridges, and they're cheaper and can have a stylus replacement at a reasonable cost. When the rest of your system gets really good, then you can take advantage of the performance improvement of a MC.