Mathematically speaking, a plane is a two-dimensional space -- infinitely broad and wide, with zero thickness. If you can't quite get your head around that go read Flatland which uniquely takes place on a literal mathematical 2d plane.
In traditional games, especially those set in fantasy worlds, the term is applied to what might more rigorously be described as (without having to go learn elementally string theory to explain) separate and parallel realms of existence, like sheets of paper (which are approximately planes) stacked on top of each other. Crossing from one plane to another is almost always possible (why put them in a setting if the PCs can't get to them?), although it is more difficult in some settings than others. Planes may correspond to one another (i.e. plane-shifting and moving ten kilometers to the east will get you to the same place as if you had moved ten kilometers east and then plane-shifted),this is in essences how Warp drives in 40k work to get around the light speed barrier, but this is not necessarily the case, especially once the planes get really weird. Sufficiently powerful characters may be able to create planes or sub-planes (often called "demi-planes") on their own.
Dungeons & Dragons
The campaign settings of the Dungeons & Dragons games (and spin-offs, like Pathfinder) are generally set on the "Prime Material Plane", which is at the center of the universe. The Prime is surrounded by other planes, which almost always include Elemental Planes corresponding to the classical elements and often includes more, such as the Positive and Negative Energy Planes, planes belonging to various gods, and the afterlives of particular alignments. Planescape probably has the most planes out of any setting.
Magic: The Gathering
The different realms of the setting of Magic: The Gathering are called Planes, but they are generally limited to a single planet in scale. Most people live their lives on the plane on which they were born, though some people are able to travel among them. Such people are called Planeswalkers, and they are the most powerful characters in the setting (and also the Player Characters).
List of Planes
- Alara, a plane that was split into five shards, each containing three colors of mana.
- Dominaria, the "hub" plane of existence at the center of Dominia (the planeswalkers' name for the multiverse).
- Innistrad, every gothic-horror movie mashed together.
- Kamigawa, bits and pieces of Japanese history and mythology.
- Lorwyn and Shadowmoor, a plane that cycles between beautiful summertime and dusky gloom.
- Mirrodin, later New Phyrexia.
- Phyrexia, a mash-up of H. R. Giger and the Borg.
- Ravnica, the City of Guilds.
- Zendikar, a plane with lots of ancient ruins, wickedly powerful mana, and bizarre landscapes. It also has eldritch abominations sleeping beneath the surface.