ZIP is probably the best-known file compression format in the world. The format was originally developed by Phil Katz, the founder of PKWARE, back in 1989. The first tool ever to use the compression method was called PKZIP, developed by Katz's company. Nowadays, .zip is almost exclusively used as the file extension for ZIP-compressed files.
Nowadays, the most popular ZIP compression software must be WinZIP, but several freeware tools exist as well, including Info-ZIP and 7-Zip.
ZIP is also used as the compression method for several other file formats as well, including OpenOffice's native document files.
In order to extract (or create) ZIP files, you need to download one of the ZIP-capable tools. Here are two good download options:
ZIP's main competitor is currently called RAR that provides slightly better compression ratio and better handling of split archives. However, support for ZIP files is so universal that without a specific need for extremely good compression ratio, ZIP should be always considered as one of the main options for file compression.
Related software tools
7-Zip (Open source)
7-Zip is a free compression software that can create and open .zip, .rar and .7z file formats, among others.
WinZip is the most popular compression/decompression utility for Windows.