Basic list of components for a computer (and abbreviations):

Motherboard (Mobo)
This controls everything in your computer, including how well it overclocks, what parts it takes and needs, and some more complicated features we’ll get into later.

Processor (CPU)
One half of the brains of your computer: There are various sub-types of processor, but the easiest way of describing it is just a brain.

Memory (RAM)
The other half of your brains, the more of it you have, the more things you can do at once, think of it as a “to do list” your computer carries around internally.

Hard drive (HDD) or Solid State Drive (SSD)
This is like the memory bank in your brain, everything your computer needs to store, run, and function is in here. HDDs and SSDs are different in that SSDs are (for the most part) MUCH faster than HDDs, though are a lot more expensive.

Optical Drive (ODD)
You put discs in it and your computer reads the data on the disc, this could be a CD, DVD, or Blu-ray (BD) drive, and can read or write. Some can actually write labels onto the disc for you, these are known as Lightscribe, but the discs are pretty expensive.

Power Supply (PSU)
This powers everything in your computer, it takes the 230V (in the UK at least) and distributes it evenly in either 3.3v, 5v, 7v or 12v voltages to the parts of the computer that need it and ground the entire thing so you don’t get a shock when you touch your case… which brings us onto:

The optional extras:

The case holds your system, contrary to popular belief you don’t NEED a case for a computer, though it’s highly recommended due to the fragile nature of computer components

Sound card
Again, not always required but people who are into their sound tend to get them to improve the quality of their sound

Graphics/Video card (GPU/GFX)
Not always needed, but highly recommended if you’re doing anything more demanding than playing Youtube videos

Heatsink and Fan (HS/F)
Highly recommended you get one of these and some decent Thermal paste/grease/pads if you’re building a computer. Though not required, aftermarket (eg. not Intel/AMD) heatsinks and fans tend to do a much better job at keeping your computer cool and working for longer. They can also be bought for Graphics cards and other, normally cooler computer parts.
Alternatively, you could just watercool your entire system (more on that later)

For cooling down your computer, a better airflow through a computer case means a cooler computer, which means it lasts longer OR that you have more overclocking* potential (again, we’ll go into that later).

Fan Controller
For controlling the speed of the fans. People tend to do this to quieten down their system when they’re not gaming heavily, it also works as a sort of hub if you don’t have enough connectors on your PSU and Mobo to accomodate all of your fans

PCI WiFi adaptor
For wireless internet. Not a lot of computers will come with one of these (though most modern laptops will). The easy solution is to get a highly breakable USB WiFi stick, though this offers more flexibility with regards to aerials and such.

PCI Network card
Most people will never ever need this, but it’s good to know. Some people that have high-speed systems or are on a high-speed network buy extra Network cards, most motherboards have a network socket plugged in, but those that need the extra ports or extra “oomph” will quite happily buy these.

Internal addon card (other)
Usually people would never need to go near these, but there are the odd few. These are made for people that have a motherboard that just isn’t enough, they might want more SATA ports to plug hard drives in, they might want a different functionality (like SCSI drives, which are still used in larger servers). Some offer external solutions as well like USB, E-SATA and Firewire connectivity, as well as freeview TV capabilities.

Common peripherals:
Mouse and Keyboard
…Let’s be honest, do I really have to explain these to you?

Without this you’d either have a HUGE paper bill, or great difficulty using a computer… The only general difference is the difference between CRT and TFT/LCD, the latter being flat and more spacesaving.

Used to listen to sound coming from your computer… But you already knew that

Printer/Scanner/All in one
Taking stuff off the computer and putting it onto paper (or vice-versa)