If we put it in simple words, turntable work by dragging a tiny diamond through the grooves of a record. The cartridge is supposed to convert the resulting vibrational energy into a nice analog sound. The quality of your cartridge has a big word on the sound quality, as it figures.
When we go into details, we see that the shape of the stylus influences how it makes contact with the record groove. The stylus is abler to track modulations in the groove if the contact radius is narrow. Common shapes of styli are conical and elliptical. Elliptical shaped styli come with smaller contact radius that the conical ones and this is why the first one’s trace more accurately grooves and extract more musical info.
You need a stiff cantilever and as light as possible for an efficient transfer of the vibrational energy from the stylus tip to the magnet. The material, size and build of the cantilever are related to the ability of the cartridge to reproduce a range of audio frequencies. You can get aluminum, carbon, boron and even copper alloys as material on the cantilever.
The ability of the stylus to track a modulated record groove translates into its track ability. Track ability depends on many factors, like stylus shape, cartridge alignment and tone arm compatibility.
When we talk about the generator types, we see the moving magnet type (MM) and moving coil type(MC). MM cartridges are the most popular ones as the MC cartridges are lower output and need a preamp with a special MC setting. This is why MC cartridges are also more expensive.
Don’t forget the mount type of the cartridge. Standard mount cartridges are secured to the tonearm by 2 vertical screws, whereas the P-mounts have four slender pins that plug directly into tonearms.