The pickup cartridge is the device mounted at the end of the tonearm which holds the stylus (usually diamond) onto the record's groove. The groove vibrates the stylus, which transfers this movement into the body of the cartridge via a short rod (known as the cantilever). The cartridge then has the task of converting the mechanical vibration into an electrical signal. Different types of cartridge use different methods to generate the electrical signal.

Magnetic Cartridges

The vast majority of cartridges use electromagnetic induction to generate the signal (and are therefore known as "magnetic cartridges"). This method involves moving a permanent magnet and a coil of wire (a pair of coils for stereo) relative to one another.

Wikipedia article on magnetic cartridges.

Moving Magnet

As the name implies, in this type the coils of wire are fixed inside the cartridge assembly, and the magnet, mounted on the cantilever, is moved relative to them. Most DJ and lower-price home audio cartridges are of this type. There are some advantages to this type:

Since there is no physical connection between the sylus and the wiring, the stylus can easily be made replaceable. Most (but not all) modern MM cartridges offer replacement stylii, at 50-80% of the cost of a new cartridge.
Since the coils of wire are fixed, they can use a large number of turns, thereby increasing the output level from the cartridge (typically around 5mV), which will then require less subsequent amplification and hence lower noise compared to moving-coil cartridges.
Moving magnet cartridges are generally much easier to manufacture and are less expensive than moving coil carts. MM models usually range from $20 to $300.

The large number of turns on the coils give moving magnet cartridges a high output impedance (typically a few kiloohms), much of which is inductive. If the cartridge outputs are wired directly into a high input impedance preamplifier, the cartridge will form an RL filter, which can lowpass the signal in the 10-20khz range and compromise performance. Furthermore, the high mass of the cantilever/magnet assembly generally causes a resonance in the high frequency audio range, although not enough to complement the coil inductance. Two elements are added to alleviate these issues.

A capacitance is added to the circuit to form an LC tank circuit. This boosts the high end response substantially, but also creates a large resonance, potentially in the audio range. Preamps usually come with 50-200pF of capacitance at the inputs, but often, the cabling from the cartridge to the preamp contains up to 100pF of capacitance.
To reduce the size of this resonance, a relatively low resistance is added between the signal and ground, to dissipate the resonant energy, smoothing out the high frequency response. This resistance should be low enough as to reduce the HF peak, but high enough so as to not compromise the HF response completely. Most cartridge manufacturers have standardized at 47kOhm as the "usual" preamp input resistance.

Moving Coil

In a moving coil cartridge, the magnet is fixed and the coils are wound onto arms at the far end of the cantilever. Therefore the stylus cause the coils to move relative to the magnet. There are usually several advantages to this scheme:

The cantilever/wire assembly is extremely light, and unlike a moving magnet, the mechanical resonance of the cantilever is generally in the ultrasonic range.
The wire assembly usually has much fewer turns of wire, compare to moving-magnet cartridges. This substantially reduces the source impedance of the cartridge and makes it much easier to drive electrically.
MC cartridges, being at the higher end of cartridge technology, tend to be designed to much tighter specifications than other cartridge technologies.

There are several disadvantages.

The cantilever assembly is usually not user replacable, as it is electrically wired to the rest of the cartridge.
MC cartridges are extremely expensive. The cheapest cartridges are usually around $200. The most expensive cartridges are over $10000.
Because of of the smaller wire assembly of MC cartridges compared to MM cartridges, MC carts tend to have vastly reduced electrical sensitivity - often 20-30db less than MM. This requires a much more advanced preamp design than in order to achieve the same noise specifications. (Note that some MC carts are designed for high output and do not suffer from this issue.)

Nevertheless, MC cartridges are usually favored for their high fidelity compared to MM cartridges.

Crystal and Ceramic Cartridges

These types of cartridge use the piezo-electric effect. The vibrations of the stylus are used to apply pressure on a crystal or block of material which has a piezo effect, which generates a voltage. The generated voltage can be quite high (up to 100mV). And unlike magnetic cartridges which generate a signal proportional to the velocity of the stylus deflection, crystal/ceramic cartridge signals are proportional to the amplitude of the stylus deflection. This effectively forms a lowpass filter of the velocity-based signal, requiring little to no additional RIAA equalization. As a result, crystal/ceramic cartridges do not require any phono preamp, are very easy to manufacture, and are still used today for very low cost turntable systems.

Crystal/ceramic cartridges have many major disadvantages. First, they tend to have much higher levels of distortion and noise compared to magnetic cartridges. Second, because the stylus/cantilever system is mechanically coupled to the cartridge, these cartridges tend to have extremely low compliance. This tends to compromise the high frequency response. Third, their purported compatibility with line level stages generally comes at the expense of any sort of close accuracy of the RIAA reproducing curve at high frequencies.

Other Types of Cartridge

Grado, a pioneer cartridge manufacturer that was the primary innovator of Moving-Magnet cartridges, calls their technology "Moving Iron". It is occasionally seen in other brands.

Cantilever

The cantilever is responsible for mechanically coupling the stylus (which actually touches the vinyl) to the electrical transducers.

It has been known to be made of several materials. The primary technical consideration in the choice of metal is its mass, followed by its mechanical deformation characteristics. These affect the cantilever assembly's various resonance characteristics.

Aluminum (most common). Sometimes solid and sometimes rolled from a thin sheet to reduce mass.
Boron
Beryllium (used in Shure V15 carts, now no longer available due to toxicity concerns during manufacturing)
Gold-plated metal (usually aluminum)

Stylus

A stylus is the piece that actually touches the vinyl and tracks its modulations. It is usually the only component of a turntable that really "goes bad" over time and must be replaced, usually along with an assembly, or the entire cartridge.

Stylii are nowadays univerally made of diamond. No other material is as hard or durable. Worn stylii develop polished facets that act as chisels on the vinyl, causing permanent signal damage. A diamond stylus will generall wear out after somewhere between 200 and 2000 hours of use. Inferior materials (sapphire, osmium) will wear out far faster - possibly within 1-50 hours.

The most important stylus characteristic is how it is cut, which determines the profile of the stylus on the groove. This measurably and audibly impacts the signal quality, the tracking force and the lifetime of the stylus. If the stylus touches a large surface area of the record, a larger tracking force is required to exert the same pressure on the surface, which also reduces the life of the stylus. If the surface area is oriented horizontally rather than vertically - if it covers more of the groove radially - then the high frequency response is compromised. On the other hand, a stylus with a very small contacting surface area tends to be very sensitive to proper alignment and tracking force, and is not as user-friendly to configure.

The simplest way to cut a stylus is spherically. A pure spherical cut is pathologically bad and is never seen for microgroove records. Rather, a stylus is cut conically, with a spherical tip. These generally have the worst frequency response and a short lifetime. However, a few high end cartridges - notably the legendary Denon DL-103 - still use a conical tip. Most DJ cartridges use conical tip stylii.

The stylus can be cut in an ellipsoidal shape, so that the contact area forms an ellipse. These are very common in audiophile cartridges.

Finally, there are a number of different cuts related to the idea of strongly eccentric, vertically oriented contact areas on the record. These include names like "fine line", "micro-line", "Shibata", etc. These are also very common for audiophile cartridges.

It is worth noting that quadraphonic systems, which require much more signal bandwidth than stereo, used fine line or Shibata stylii in the 1970s.