In fantasy role-playing games, a plane (from plane of existence) is a self-contained pocket of reality. It can contain anything from a world, a universe, an amount of matter or energy or other things still. They can be either limited in scale or infinite in size: some are small enough to walk across in minutes, others can be traveled months by foot, others required dedicated vessels to traverse but are impossible to do so in a human lifetime, and others are infinite in size.
What they have in common is that while they can border on one another they are self-contained: while it is possible to walk between them in some places they are distinct parts of reality. The concept differs from that of a Multiverse: planes are distinct from one another and while they can share some similarities, they are easily discernable from one another.
When people talk about "the planes" they almost always talk about the planes as used by Dungeons & Dragons, specifically by the Planescape campaign setting. They contain a series of Material Planes (whom contain several worlds used in various D&D settings such as Greyhawk, Forgotten Realms, Dragonlance and more), a series of demi-planes made by various inhabitants of the Material and others, the Inner Planes who compose of the matter and energy on which reality is build (air, earth, fire, water and two energy planes: positive and negative), the Astral Sea (the realm of dreams, thoughts, and psionics), the Outer Planes (a variety of planes all of infinite size, many of whom based on depictions of some form of afterlife, including Dante's depictions of heaven and hell, afterlives based on Greek and Norse mythology, a plane based on the Happy Hunting Ground of Native American tribes of the Great Plains and more. In the center lies the city of Sigil, which contains portals (ways to teleport form Sigil to somewhere else) to many places in reality. Finally there is the Far Realm, based on the works of H.P. Lovecrafts and others like him; a plane beyond planes of sorts where beings alien to reality preside and tend to have malevolent intend towards the inhabitants of reality.
Games set on the planes rather than just one material plane tend to be grander in scope: it deals with meeting beings more alien (and often dangerous, though not always malevolent) than those on the native plane of the players, and can deal with a variety of subjects such as interplanar adventuring, war between good and bad aligned planes, threats to the gods or invasion from the Far Realm.