Cartridges, broadly speaking, come in two types -
Moving Magnet [MM] and Moving Coil [MC]
Both types can come pre-fitted & balanced, as part of a pre-built system, or bought as separates & require the user to balance.
Balancing requires specialist equipment, so if that's your choice, allow an extra $100 or so to be able to do that at home.
Moving Magnet [MM]
MM is the simplest & cheapest to make & maintain & all amps with a phono input will support it.
A 'phono' input has a special preamplifier, known as an RIAA preamp [details outside this scope] which applies a specific equalisation [EQ] curve to the input - matching, yet opposite to - the curve the the record was cut using.
MM cartridges have a stylus (needle) you can change yourself, after 600-800 hours of use.
The sound of MM cartridges is described as 'warm'.
This doesn't necessarily mean muddy or indistinct, but there is more weight to move than MC & this can result in less clarity & transparency in the high frequencies.
Moving Coil [MC]
MC is more expensive than MM & is not supported by all amps. It has a lower output volume/voltage than MM & therefore requires an extra preamp stage before the RIAA preamp - which increases the volume only, & does not affect the EQ. This adds to the amp cost, of course.
Some MC cartridges actually have an even lower designed output, which requires an even more powerful & expensive preamp.
MC styli cannot be changed by the user & the cartridge must be sent back to the maker for replacement. They last the same length of time as an MM stylus.
The characteristics of an MC cartridge are described as 'clear' and 'open'.
Needing less tracking pressure than MM, they exert less downward & drag force on the vinyl & are said to increase the life of the record.
To add to the confusion, there are variations on MM, called Moving Iron [MI] & Moving Micro Cross [usually just called moving cross, rather than MMC, as it would get really confusing otherwise]
MI is very similar to MM, but has an iron core instead of the magnet. Sonically it is described as 'somewhere between MM & MC'
Moving cross is proprietary to Bang & Olufsen & so is not a consideration unless that is your target system.
Pick one that fits both your budget and your ear.
If you can't tell the difference in a blind test, buy the cheaper one.
If your amp doesn't support MC, your choice is down to MM (MI is relatively rare.)