How can I tell what kind of hardware I have?

There are several different places to check, depending on what you're looking for.

PC

System Properties

How to get there: Right-click on My Computer and select Properties. The General tab should come up by default.
Hardware information: Processor, RAM. Also tells you what version of Windows you're running and who it's registered to.

Device Manager

How to get there: Right-click on My Computer and select Manage. Choose Device Manager from the left sidebar.
Hardware information: Peripherals, type of processor, type of hard drive (but not size).

Disk Management

How to get there: Right-click on My Computer and select Manage. Chose Disk Management from the left sidebar.
Hardware information: Sizes of your hard drive(s) and all partitions.

Mac

The Apple System Profiler, which can be found by going to Apple > About This Mac and selecting More Info....

System Profiler provides a lot of information about your computer. Probably more than you actually needed when you were looking for information. It does give you a basic overview of the specs of your computer (including the computer's serial number).

It also provides a complete breakdown of the hardware in your system, from ATA (which includes hard drives and CD/DVD drives) to Airport to Memory (through System Profiler, you can determine the exact specs of your RAM) to information about any USB devices you have connected to your computer.