A computer's hardware refers to the electrical parts and components inside a computer which power the system. The operating system and other software are then installed upon - and make use of - the hardware.
For example, a hard-drive is a piece of computer hardware which stores data such as the operating system, software, files, pictures, music etc. The CPU (central processing unit, also called processor) is the 'brains' of the computer - it is responsible for carrying out the various instructions and computations of the system.
And the RAM - which stands for Random Access Memory - temporarily stores data generated by the system (for example, if you are typing a letter but haven't saved it yet, the letter data is stored in the RAM).
Once the system is turned off, all the data held in the RAM is erased (unlike the hard-drive which stores the data even after the computer is switched off).
These are just three such computer hardware parts. Other common hardware components include the following:
Motherboard - This is the large square or rectangle component which all the other hardware parts connect to (hence its name, i.e. the mother-board)
Power Supply Unit (PSU) - This is where the power cable (from the wall socket) goes into, and it then converts that electricity and disperses it to the various components in the system
Optical Disk Drive - This is where you insert CDs or DVDs to play/burn them
Graphics Card - The graphics card is a powerful component which generates and outputs images to the computer monitor. Not all systems have these, although all gaming computers tend to have one
Sound Card - A sound card processes, generates and outputs audio. Like graphics cards, not all computers have these, although systems which are built to create and edit audio (for example music) tend to have one
Case - The case houses all of the above computer components
The picture below shows a custom built computer system which has a motherboard, CPU, RAM, hard-drive, PSU, optical (CD/DVD) drive, graphics card and a case. Some of the components are hard to see or are obscured (such as the CPU which always has a large heatsink/cooler over it due to large heat output), although you can see most of the components in it:
(Sorry about all the cables!) A picture of this same system from another perspective (showing the buttons at the front of a computer) with the case off the front can be seen below:
As can be seen in the following list, this category has various sections relating to different computer hardware. It also covers computer cases, which house the various hardware components. If you are planning on building or customizing your computer, this is the section to read!