From Bulbapedia, the community-driven Pokémon encyclopedia.
A legendary duo is a pair of closely related Pokémon, either representing teamwork or opposition. Much like how every region has a legendary trio, every region also has a legendary duo. In the games, they are often found after the player gets past the Elite Four but often aren't as important in the main storyline as the trios are. Duos are featured in several of the movies, either in their respective pairs, or on their own, as a keeper of peace. Sometimes pairs of Pokémon once thought to be legendary duos are revealed to actually be legendary trios upon the release of the third version. A primary example would be Dialga and Palkia.
Every generation thus far has featured at least one legendary duo. Duos introduced in even generations tend to be opposites, like Lugia and Ho-Oh. Conversely, duos introduced in odd numbered generations tend to be more closely related, such as Latios and Latias.
The Mew duo
Mewtwo and Mew make up Kanto’s legendary duo. They are both psychic type feline-like Pokémon that are very elusive. Mewtwo was cloned from Mew, meaning the two share DNA and abilities.
In the first Pokémon movie, Mew embodied innocence and joy, but Mewtwo's experiences as he gained consciousness gave him a jaded and cynical view of humanity. The duality expressed between the two characters formed the basis of their conflict, but innocent Mew, with the help of Ash Ketchum, is able to sway Mewtwo's contempt.
The Tower duo
Lugia and Ho-Oh in promotional art for the TCG
The tower duo is the legendary duo of Johto that consists of Lugia and Ho-Oh . Lugia and Ho-Oh are often regarded as polar opposites. Lugia represents the moon, sea, storms, the color silver and is the mascot of Pokémon Silver and Pokémon SoulSilver, while Ho-Oh represents the sun, sky, rainbows, the color gold, and is the mascot of Pokémon Gold and Pokémon HeartGold. Both are also trio masters; Lugia is the master of the legendary birds, while Ho-Oh is the master of the legendary beasts.
Their legend is introduced in Generation II, and states that the two birds each resided on a tower: Lugia on the Brass Tower, and Ho-Oh on the Tin Tower. When the Brass Tower was destroyed in Johto (prior to the start of the second generation games), the pair was split up thus creating the legendary beasts: Entei, Raikou, and Suicune.
The Eon duo
The eon duo is a legendary duo of Hoenn that consists of Latias and Latios (sometimes collectively called Lati@s). They are region's roaming Pokémon and they share their typing, four of their base stats and even several moves they learn when they level up. Both were featured in the movie Pokémon Heroes, where they were revealed to be playful siblings.
In the Generation III games, one can be found roaming Hoenn, depending on the player's earlier actions (see Eon Pokémon). Also, in the Generation IV remakes of Pokémon Gold, Latias (in HeartGold) and Latios (in SoulSilver) are found roaming Kanto after speaking to Steven Stone. Also, if the player has access to the Mysterious Crystal, then the oppisite game version may encounter the other twin.
The Lunar duo
The lunar duo is a legendary duo of Sinnoh that consists of Cresselia and Darkrai. Cresselia and Darkrai are commonly seen as polar opposites, just like the aforementioned Johto bird duo of Lugia and Ho-Oh. Cresselia and Darkrai are also known as the subconscious duo because of how Darkrai is said to give people nightmares, while Cresselia does the opposite (giving people pleasant dreams). Furthermore, Cresselia and Darkrai also appear in Pokémon Mystery Dungeon 2 as enemies. They made their anime debuts in Sleepless in Pre-Battle! where Cresselia battled Darkrai.