There are many components inside the computer that can be upgraded.
Graphics Card: The graphics card inside a computer can allow for different resolutions, improved graphical processing power, and increased pixel display capability.
Motherboard: Upgrading your mother board can allow for more space for CPU fans, additional input slots(USB, firewire, etc.), compatibility with CPU's, and extra input slots for multiple hard drives.
RAM(Random Access Memory): Having more RAM will result in your computer "thinking" quicker and being able to handle heavier data loads more smoothly. You will be able to run applications and games with less lag, or choppiness.
Hard Drive: Buying a new hard drive can provide many benefits. You can increase your data storage capabilities on a cheap budget by opting for a HDD(hard disk drive), they are cheaper than SSD's(solid state drive) but not as fast and are susceptible to crashing. SSD's on the other hand offer increased speed and no chance of crashing(due to the absence of moving parts inside), but at a much heftier price.
Cooling System: There are advanced cooling systems that utilize water, but the majority of traditional computer cooling comes from fans. Mini fans can be purchased and added in the amount determined by whichever case you purchase. Your motherboard also has a limited amount of inputs for fans, so it may need to be upgraded as well.
Tower(Case computer is in): Choosing an appropriate tower is very important because computer components all vary in size. You may need a large case just for a large motherboard to fit, or you might be designing a computer to minimize clutter and therefore opt for a smaller tower. You need room for air to flow though, so parts don't overheat and break. Keeping your computer cool is part of properly maintaining it.