A planeswalker is someone who travels from one plane of existence to another. They appear in the two most popular works of Wizards of the Coast, namely Magic: The Gathering and Dungeons and Dragons, specifically Planescape.
In the Cant of Planescape, a planeswalker is any old cutter who travels between the planes. It doesn't matter whether they achieve this by using the portals or through their own magic: all those who travel from one plane to another are worthy of the title of planeswalkers.
Planeswalkers are distinguished by a Spark that manifests at some point during their lives, during either a very dramatic ordeal. This manifestation usually involves the newly-minted planeswalker involuntarily jumping to another plane. Not many planeswalkers survive this first jump, but the ones who have the right combination of survival skills and good luck to make it past the first few weeks are not to be underestimated. Only one in a million people are born with the Spark, and of those only one in a million have their Spark ignite during their lives. This makes them incredibly rare, and as such few non-planeswalkers know about them.
Originally, Planeswalkers were effectively gods; they could transport beings from across the multiverse as they saw fit, create portals and full-fledged planes, they could take any form they liked, could speak any language, and they were virtually immortal. They didn't age in the slightest, and unless you managed to strike what they "believed" was their most valuable organ, any damage you did to them could be almost instantly undone; planeswalkers of the original stripe (called "Oldwalkers" by fans) were a central consciousness and their physical forms were energy projections shaped by their own will. This tended to have bad effects on their sanity, and most came to regard non-Planeswalkers as inherently insignificant.
But... then came "The Mending". Long story short, the multiverse was threatening to collapse as Dominaria began to full apart due to temporal rifts, which were spreading out into other worlds. Finally, a planeswaker named Jeska sacrificed her life to close the central rift, healing all of the temporal rifts throughout the multiverse, but somehow fundamentally altering the planeswalker's Spark in the process. "Nuwalkers", as they are nicknamed, are still powerful mages, but that stems from the opportunities that planeswalking grants to find new locales to draw mana from and new spells to learn; there are far fewer inherent bonuses that come from the Spark itself. They age at a normal rate, and they are as squishy as any normal mortal. They can no longer make planes, and transporting non-planeswalkers became completely impossible. This caused great chaos amongst the Oldwalkers who were alive after the Mending was complete, as they found their powers drastically altered. In particular, Nicol Bolas went on the rampage, trying to come up with plots to restore his own Spark to its former potency, whilst Liliana Vess was driven into bargaining with demons in a desperate attempt to regain her former eternal youth.
In games of Magic: The Gathering, the players themselves are planeswalkers. Players' decks ("Libraries") represent all of the spells they know, all the creatures they can summon, and all of the places whose natural energy (or mana) they can tap. Many cards involving shuffling or searching or otherwise manipulating hands and decks are flavored as being some kind of mental manipulation -- trying to recall how to summon a particular desired creature, or trying to scramble or erase the opponent's knowledge of a particular spell.
In a move that would make Xzibit proud Wizards of the Coast began to make planeswalker cards to allow players to play planeswalkers while being planeswalkers. The flavorful explanation for this is that the player is calling on the help of an allied planeswalker; the related mechanics somewhat reflect this, as planeswalker cards have their own selection of abilities they can use on their own akin to a player casting spells, and each has a pool of "loyalty counters" that are exhausted through taking damage and firing off their more powerful abilities. In addition, planeswalkers can be attacked by an opponents creatures, just like players (creatures can't attack anything else). Once all the counters are exhausted the card is discarded, representing that planeswalker's literal loyalty to the player being exhausted and them leaving off to find something better to do.
|Planeswalkers of Magic: The Gathering|
|Original Five:|| Ajani Goldmane - Chandra Nalaar|
Garruk Wildspeaker - Jace Beleren - Liliana Vess
|Alara:||Elspeth Tirel - Nicol Bolas - Sarkhan Vol - Tezzeret|
|Zendikar:||Gideon Jura - Nissa Revane - Sorin Markov|
|Scars of Mirrodin:||Karn - Koth of the Hammer - Venser|
|Innistrad:||Tamiyo - Tibalt|
|Return to Ravnica:||Domri Rade - Ral Zarek - Vraska|
|Theros:||Ashiok - Kiora - Xenagos|
|Tarkir:||Ugin - Narset|
|Kaladesh:||Dovin Baan - Saheeli Rai|
|Other:||Dack Fayden - Vivien Reid|
|Commander 2014:||Daretti - Freyalise - Nahiri - Ob Nixilis - Teferi|
|Pre-mending:||Bo Levar - Commodore Guff - Jaya Ballard - Urza|
In Plane Shift
When WoTC began releasing their Plane Shift articles, which attempt open up the worlds of MtG for play in Dungeons & Dragons 5th Edition, it naturally came time for them to discuss how to convert being a planeswalker to D&D mechanics. In Plane Shift: Amonkhet, they did just that, and recommended that it basically become a background element, mechanically no different from the overland travel; after all, in a planewalking campaign, travel between worlds is a story function, not a rules function, or at least that's their argument. The only mechanical suggestions were that it takes a minute-long period of meditation/focusing for the planeswalker, akin to performing a ritual, and that this usually can't be done in combat. To represent the "reflexive planeshift" aspect of Planeswalkers, they can make a Charisma saving throw when hit by an attack that would drop them to 0 hit points, with a DC equal to the damage inflicted; if they pass this save, they teleport away to a random plane instead of taking damage.