The Legendary Pokemon, as of Gen 7, are Articuno, Zapdos, Moltres, Mewtwo, Mew, Raikou, Entei, Suicune, Lugia, Ho-oh, Celebi, Regirock, Regice, Registeel, Latias, Latios, Kyogre, Groudon, Rayquaza, Jirachi, Deoxys, Mesprit, Uxie, Azelf, Dialga, Palkia, Heatran, Regigigas, Giratina, Cresselia, Phione, Manaphy, Darkrai, Shaymin, Arceus, Victini, Cobalion, Terrakion, Virizion, Tornadus, Thundurus, Reshiram, Zekrom, Landorus, Kyurem, Keldeo, Meloetta, Genesect, Xerneas, Yveltal, Zygarde, Diancie, Hoopa, Volcanion, Tapu Koko, Tapu Lele, Tapu Bulu, Tapu Fini, Cosmog, Cosmoem, Solgaleo, Lunala, Necrozma, Magearna, Marshadow, and Zeraora.
There are also Mythical Pokemon, which are basically event-only legendary Pokemon. (I personally consider Mythical Pokemon to be a type of legendary Pokemon, but some people consider them to be separate) The mythical Pokemon are Mew, Celebi, Jirachi, Deoxys, Manaphy, Phione, Darkrai, Shaymin, Arceus, Victini, Keldeo, Meloetta, Diancie, Hoopa, Volcanion, Magearna, Marshadow, and Zeraora.
Phione and Silvally are considered legendary by some people, and their status as legendary has always been controversial.
Characteristics of Legendary Pokemon:
- All legendary Pokemon (excluding Manaphy) cannot breed.
- Many legendary Pokemon (excluding Latios, Latias, Heatran, Cresselia, Tornadus, Thundurus, and Landorus) are also genderless.
- All legendary Pokemon, excluding Cosmog, Cosmoem, Solgaleo, and Lunala, do not have evolutionary families.
- Quite a few battles with legendary Pokemon are initiated through interacting with a static sprite.
- Since Gen 2, every Pokemon game, excluding FireRed and LeafGreen has a legendary Pokemon that is very significant to the game's story on the game's boxart.
- Legendary Pokemon can often only be caught once in each game.
- Legendary Pokemon often have stats higher than many other Pokemon (at least 580, except the Tapus for some reason)
- Although this isn't specific to Legendary Pokemon, some Legendary Pokemon have signature moves and/or abilities that only that Pokemon can have.
Sources: This, this, and this.