The cartridge or the pickup is one of the most critical components in a turntable and probably the one component that has the most impact on sound quality.
The cartridge should if possible be evaluated or considered together with the specific arm it is going to be used with. Without this knowledge, it can be difficult to fairly judge a cartridge. The two work in symbios and can almost be regarded as ONE system. The reasons for this is that different arms are compatible with different cartridges. For example, a long heavy 12" arm will not work well with high compliance cartridges and the system will be very sensitive to resonance. Such an arm will only work well with a low compliance cart. The main problem is that the combination will always have a resonance frequency though. If the resonance frequency is too low there will very likely be noticeable interference that could be heard. As a rule of thumb (with some exceptions of course), an optimal resonance frequency should be at least 10Hz or higher, but not over 14-15Hz for optimal results. The subsonic resonance frequency is not in the audible range but a too low resonance will have effects far up in the frequency spectrum, degrading the sound quality. So, to conclude; a long heavy most likely works best with a low compliance cart and a short light arm with a higher compliance cart. But this is of course not the only consideration, the weight of the cartridge also plays a significant role. A heavy cartridge will most likely not be optimal in a light arm. A light cartridge might sometimes work sufficiently in a heavier arm though, especially if a heavier headshell is used.

The classic Ortofon SPU MC cartridge which was the world's first MC cartridge. This is a Gold model with Elliptical stylus. A Fantastic cartridge but very tricky to use because of its weight: 32gr - a heavy ultra-stable and curved arm is a prerequisite so very few arms can even be considered. Most arms, even S-curved medium mass ones can't even balance out such a high mass. Of course the arm has to use an SME style/chuck type headshell connector as the SPU is a complete utnit with integral headshell.
COMPLIANCE
So what is COMPLIANCE? Well, simply put, it is the stiffness of the needle suspension. A high compliance cartrige is softly suspended and a low compliance cart has a stiff suspension. Hence, the compliance plays a vital role in the total resonance of the cart/arm system. Usually, a high compliance cart will also track best. However a low compliance cartridge can also be a very good tracker! Confusing - yes! In the world of Turntables not all things are what they seem . . . .
TYPES OF CARTRIDGES
Then there are of course different types af cartridges. The two most common are the MM - Moving Magnet and the MC Moving Coil types. These two types probably cover 85-90% of all carts manufactures but there are many other types and variations also. Simplified, these two types can be regarded as oposites. The MM type uses a Moving Magnet fixed to the cantilever and fixed coils and the MC type uses Moving coils fixed to the cantilver and fixed magnet.
A common notion is that MC carts are always better than MM carts. However, the reality is not that simple. A well designed MM cart can be superior to an MC one. This misunderstanding could have its roots in the fact that many Moving Coil cartridges are more expensive than their MM counterparts. The main reson for this is that MC cartridges are often more difficult and complicated to manufacture. Another disadvantage with MC cartridges is that they usually have poorer traction than MM counterparts. However, the fact that many of the cartridges considered to be the best you can buy are Moving Coil carts is difficult to ignore.

This is my rarest cartridge, a Win Lab SDT 10-2 Boron Whisker. It uses a semi-conductor to reproduce the signals and has no coils nor magnets. Requires a special Pre-Amp with a BIAS VOLTAGE to work. It is a an AMPLITUDE generating design and not VELOCITY generating as all magnetic MM or MC cartridges. It's one of the best cartridges I've heard - sounds very much like the Ortofon SPU Gold above, but even more detailed and crystal clear - incredibly detailed reproduction with razor sharp resolution. Uses a long pure and solid BORON cantilever.
THE STYLUS
Then there is of course the matter of the actual stylus tip which tracks the records. This tip is made of DIAMOND and can be CUT in many different ways. The most common types are the Conical cut and the Elliptical cut. These two types probably covers 85-90% or more of all cartridges produced. Cartridges with these cuts can be very good and have faithful reproduction. However, in the quest for even better tracking, engineers came up with more advanced cuts like the Shibata (JVC) cut, the Line-Contact cut (very similar), the Micro Ridge and other types. The more advanced cuts will track the grooves of the vinyl more accurately and also reduce friction and thus record wear. However, these advance cuts are very complex and difficult to cut and thus more expensive to manufacture.

A microscope picture of another MC classic - the venereable Denon DL103. It uses a solid nude diamond with a spherical cut. This is a very unusual design where the diamond is mounted inside the cantilever tube.
THE CANTILEVER
Then there is the needlearm which is called the cantilever. It can be manufactured of a variety of materials, from aluminium to high-tech materials like Boron, Beryllium or even Ruby or pure Diamond (!) like some Dynavector carts. In general, the lighter and stronger/stiffer the needle arm, the better. The shape of the cantilever can also be varied. A conical arm is more rigid but also more expensive.