Read only memory or ROM is used as a storage medium in computers. Because it cannot (easily) be written to, its main uses lie in the distribution of software that is very closely related to a certain hardware, and not likely to need upgrading. For example a graphics card may implement some basic functionality through software contained on a ROM.
There is a trend to put less and less software into static ROMs, and more on disk storage, making changes easier. Home computers in the 1980s came with their complete operating system in ROM. Upgrading to a newer version meant taking a soldering iron, and replacing the old ROM chip with a new one. By the 2000s operating system are not generally on ROM anymore. They may still rely on some software in ROMs, but even that is more likely to reside on a Flash-ROM (see below). Mobile phones are likely to have software in ROM (or at least flash ROM). Some types of video game consoles use ROM based software: the SNES, the N64, and the Game Boy. Such ROMs are sealed into plastic cases suitable for handling and repeated insertion, known as cartridges or "carts" (or Game Pak if you are Nintendo). By extension ROM may also be used in refrence to a data file that contains an image of the software normally distributed in a ROM, such as a copy of a video game cartridge (often a violation of copyright).
One reason why some data still sits in ROMs is speed -- disks are an order of magnitude slower. Even more important, though, is that you cannot read software that is needed to drive a disk from the disk itself -- see bootstrapping. Hence the BIOS or a bootloader for a computer is often on ROM.
Classic ROM chips are written to during production and cannot change content afterwards. EPROMs (Electrically Programable Read Only Memory) can be written to (programmed) via a special device, an EPROM programmer. Flash-ROMs or EEPROMs (Electrically Erasable Read Only Memory) can be written to (flashed) without taking them out of the computer. Flashing is much slower than writing to RAM (Random Access Memory) (or reading from any ROM).
RAM can be read faster than most ROMs, therefore ROM content that is used often is sometimes copied to RAM (shadowed).