There are lots of different kinds of printers, but when it comes to the ones that are most common for hobbyists and home printers, there are basically extrusion printers and resin printers. Extrusion printers are also called “FDM Printers”, and resin printers...well, they have a lot of names(SLA, MSLA, LCD, DLP...). There are some differences in the technologies that specific resin printers might use, so the distinctions can be meaningful, but in the end they all pretty much work the same. To try to keep things simple, The most accepted term is "SLA," but I’m just going to call them resin printers.
FDM stands for “Fused Deposition Modeling,” which is a fancy way of saying this kind of printer works by melting layers of plastic together (although the general principles can be used to make 3D objects out of pretty much anything you can extrude and stack...chocolate, cement, ceramic, glass, meat....) These printers work like a giant pen controlled by motors to move it in 3 directions (dimensions.) Instead of leaving a trail of ink, the printer leaves a thin trail of melted plastic that hardens almost as soon as it comes out. The material is fed in from a big spool of “plastic string”, or filament, which looks like something that could be used to edge your driveway. Maybe it can. I dunno.
For most of the FDM printers you might use at home, the build volume (the largest possible size that a printer can print in all 3 directions) ranges from a cube that’s a few inches in each direction, going up to a cube that’s about 20” all around.
Some of the most common FDM materials people use are PLA, PETG, TPU, and ABS (which are abbreviations of their chemical description so if I told you what those stood for, you wouldn't care.) The materials have different pros and cons, but PLA is a favorite for being cheap, plentiful, earth-friendly (non-toxic, renewable, recyclable and technically biodegradable), and overall probably the easiest option to work with. Depending on the material used, FDM prints can be surprisingly tough or flexible like rubber.
Current resin printers work in a totally different way than FDM printers. Instead of “printing by drawing”, it’s more like “printing by shining.” Instead of plastic, the "building material" is a resin that has the consistency of syrup and is photosensitive (it becomes a solid when its exposed to certain wavelengths of light). The vat that holds the resin has a transparent bottom, and underneath it is a digital LCD “mask” screen (sort of like a cell phone screen) and a UV light. To print a layer, the printer turns on the UV light, and turns off specific pixels on the mask to let light shine through and harden the resin in the correct places on the current layer, sort of like making inverted shadow puppets. This means an entire layer can be printed at one time, and it typically only takes a few seconds per layer.
*I will now slightly complicate my over-simplification and point out that not all printers that use resin actually print this way. For decades there have been resin based 3D printers that used a laser to draw out the layer instead of this new method using an LCD screen. These laser models still exist, but the LCD-based printers are faster and cheaper and have basically the same quality, so anyone starting in resin printing will probably buy one of those. Having explained this, I have now convinced myself that I am justified in ignoring the laser-y kind, and we can move on.
The level of detail these printers can achieve is far greater than FDM printers, but the potential print size is typically much smaller (the average starter printer has a max print size of around 5” x 3” x 6”). Larger sizes are coming out, but they are more expensive. Different resins are available with different properties (like colors, flexibility and texture). The average resin makes prints that are harder than most standard FDM prints, but are typically more brittle (they don't flex, they break). The resin itself is also kind of a nasty substance, so you need to wear gloves when working with any resin that has not been cured (hardened), which is most of the time you're working with it. The resin is also messy and can put off fumes. Whenever a print is finished, it typically needs to be rinsed in isopropyl alcohol, and then final cured with a UV lamp or sunlight. Because of the nature of the resin, I find it to be a more high-maintenance process, but it definitely excels in some categories. And I also know some people think their resin printers are less frustrating than their FDM printers so they do seem to be good go-to printers for some people.