Base - That’s the back. Sometimes the PCB sits in there (like the Universal kits), sometimes it just floats (like the skeleton kits).
Hot Swap PCB - is a PCB that doesn’t need the switches soldered, switches can be added and removed without soldering. (see PCB for more)
PCB - the circuit board that links each key press to the USB. (see Programmable PCB or Hot Swap PCB for more)
Programmable PCB - this refers to a PCB that has a programmable chip on it. This way, each key can be remapped, programmed or made into a macro.
Plate - Plate is the top. It’s the part with the switch cutouts. (see universal plates for more)
Keycaps - the plastic piece that your finger touches.
Keys - are the results of all the parts. People often use this to describe a single part, when they mean switches, keycaps or pcb traces.
Soldering. You must solder the keys on to the PCB? Solder the switches to the PCB, yes.
Skeleton kit / Sandwich kit - Keyboard kits made of just two plates. 1upkeyboards is famous for this post-modern design.
Standoffs - the little pieces that go between the base (backplate) and the top plate.
Switches - these are the parts that go under the keycap and into the PCB. This is where you get to customize the feel of your keyboard. Here's a list of every switch ever made.
Universal plates - This is a plate that’s cut out to support multiple layouts.