Leviathan - He has command of the water, often attacking with tidal waves. In FF4, he's the King of the Summoned Monsters, with Ashura as his queen. You may recognize his name from the Book of Job or related Jewish mythology, in which he's, well, a great sea monster. According to the Talmud, Yahweh spends three hours of every day in the maintenance of Leviathan. His FF portrayal makes him look like a Japanese water dragon.
Asura - It looks like she actually only appears as a summon in FF4, but since that game makes her Queen of the Monsters, I think she's worth including. Unlike her draconian husband, her form is human, but she does have three different faces. She specializes in healing spells, using a different one depending on which face she's currently wearing. I'm not entirely sure which mythological tradition her name is referencing, because variations on the word were common in Indo-European languages. The most obvious source would probably be the demons of Hindu mythology, but the word is also quite possibly linked to the Avestan "ahura" (as in the name of Ahura-Mazda, the good god of Zoroastrianism), and maybe even to the Norse "Aesir." Then there's the Assyrian god Assur or Ashur, and the Semitic goddesses Asherah and Astarte. Asherah is frequently referenced in the Bible in the context of the poles that were set up as shrines to her, and artifacts from before Israel became strictly monotheistic indicate that she might have been seen as Yahweh's own consort at one point.
Bahamut - Many ancient cosmologies had the planet supported by gigantic animals, and one of these in Arabian mythology was Bahamut (a name possibly linked to the Biblical Behemoth, which also appears in the FF games), a creature that swims in the great celestial ocean. It's not entirely clear what kind of animal Bahamut was supposed to be, but some traditions say he's a fish, while others give him the head of an elephant or a hippopotamus. In Dungeons & Dragons, however, the name was used for the ruler of the good dragons, and it's in this context that the character made it into the FF series. After a brief appearance as the dragon who changes the Light Warriors' classes in FF1, he usually shows up as one of the most powerful summoned monsters in later games, using the Mega Flare attack against enemies.
Ramuh - I'm not sure how this weird transliteration came to be the standard English version of this old man's name, but he's a thunder god who uses lightning-based attacks. The most likely source for the name is Rama, Vishnu's warrior avatar in Hindu mythology, but I don't know of any particular connection between him and lightning. The name for this character in the original English translation of FF4 was Indra, who actually WAS a Hindu storm god, and hence a more suitable name.
Shiva - Another character from Hindu mythology, but this name might be even more inappropriately applied than that of Rama. In the Hindu pantheon, Shiva is the destructive aspect of the Trimurti, and is viewed as male. In the FF games, Shiva is an ice goddess.
Ifrit - Known as "Jinn" in the American FF4 (or, more accurately at the time, FF2), an ifrit is actually a kind of jinn in Arabian lore, so I suppose either name works. As an ifrit can be controlled with magic, it's an obvious choice for a summoned monster. As Ramuh's element is lightning and Shiva's ice, Ifrit's is fire.
Chocobo - This bird is primarily known as the main source of transportation in the FF games (inspired, from what I've heard, by the riding birds in Hayao Miyazaki's Nausicaä of the Valley of the Wind, but I have neither seen that movie nor read the manga), but there are several different variations. The earlier games have white species that can restore magic power and black ones that can fly, although this color-coding isn't always consistent from one game to another. Another frequently appearing type is the Fat Chocobo, a huge bird that must have an steel-lined stomach and an incredibly slow digestive system, as it can be used for storing and then retrieving excess items, including weapons and armor. Chocobos can often be summoned into battle to fight monsters, and some games occasionally have a Fat Chocobo drop on enemies and crush them.
Titan - I'm sure we all know that Titans are the giant ancestors of the Olympian gods in Greek mythology, and the Titans in the FF games maintain the huge size of their namesakes. They also eat rubies, if FF1 is any indication. Their attacks generally consist of shaking the ground.
Odin - The leader of the Norse gods tends to be one of the more powerful summons, having some attacks that will kill an enemy in one hit. In FF6, there's a way to turn Odin into Raiden, who's named after a Japanese thunder god. Since Odin isn't a thunder god in either the Norse tales or the FF games, I'm not sure why that's the name of his improved form, but there you go.
Phoenix - The mythological bird that can come back from the dead is able to extend that ability to others in the FF games. Even in games where the Phoenix summon itself doesn't appear, Phoenix Down is the standard way to bring a fallen party member back to life.