Short version: you put a hard drive in it, and plug it into your computer.
An external drive is useful to give you more storage on a laptop, which can usually only hold one drive (maybe two), and those have to be 2.5" drives.
An external drive is useful when you want to move a lot of data between computers. USB flash keys are good for moving a few gigabytes, but an external hard drive is the way to go to move hundreds of gigabytes.
An external drive is useful for backup - you attach the drive, backup your files, then detach the drive and store it somewhere safe.
Different kinds of enclosures use different interfaces. The most common is USB2, but they also come in USB3, Firewire, and eSATA. There are even a few that use some kind of SCSI, and even Ethernet.
If your enclosure is USB2, you can plug it into almost any computer. Windows has had drivers for USB2 external drives since Windows 95 SP1. Other interfaces may require drivers to be installed. Drivers for most interfaces are included in Windows 7.
A 3.5" enclosure can hold a 3.5" drive, or a 2.5" drive. If you have a 3.5" drive in it, you will have to plug in power (typically a wall-wart) to power up the drive. If you have a 2.5" drive, you may be able to power it from the USB port (depends on the hard drive, and how much power it requires, and whether the enclosure was designed to allow USB-powered operation).
Installing the drive is straightforward. You open the enclosure, connect the drive to the interface board, probably fasten the drive to the enclosure in some way, and close up the enclosure again. Be careful not to zap the interface board or the hard drive board - static can destroy either (ground yourself before opening the drive envelope, and before touching either).