The parts you'll need:
A CPU (Central processing unit)
A GPU (Graphics card, graphics processing unit) - This is what does 3D stuff, it's important for games
RAM (Random access memory) - Like short term memory in your brain
A motherboard - Basically, this is what ties it all together
Hard drives/ SSDs - They store things
A PSU (Power supply unit) - This converts the power from the wall into power which is usable for your computer
A case - This holds all of your parts
A DVD drive - Most people install their operating system via one
An OS (Operating system) - Windows, OSX, Linux etc. are examples
The GPU, or graphics card, is the most important part of any gaming system. It allows for better framerates, higher resolutions and better maximum detail settings. For a gaming system, it is therefore very important to invest a large part of one's budget into such. The CPU, or central processing unit, (processor) does not largely affect game performance. It is therefore not important to have a high-end processor such as an intel i7, as even weaker processors such as an intel i3 or i5 will not bottleneck 95% of GPUs.
For RAM, you'll want 8GB. You'll want enough hard drive space for games, and also, if you want, an SSD (Solid State Drive) as a boot drive. Solid state drives are small and expensive, but extremely fast. Put your operating system on an SSD, and your games on a regular hard drive if you decide to do so - you'll start up and shut down extremely quickly, but still have enough space.
Hope this helps!