In the anime, Professor Oak states that Articuno and Moltres combined powers of ice and fire are the source and cradle of the oceans, and that Zapdos' electric energy creates the currents and streams, which is why the three ancient creatures are closely bound to the guardian of the ocean, Lugia.
Mew: life. Considering that it looks like a foetus, Mew is the embodiment of the beginning of Pokemon life, being the Pokemon that contains the DNA of all other Pokemon.
Mew2: human interference. Mew2's appearance is the result of humans tampering with its DNA, and its violent temper is the result of being experimented on and treated like a tool, though it's finally calmed by Mew (& Ash).
the beasts find their master in Ho-Oh, who revived them when Ecruteak City's Brass Tower burned. They are said to embody the three events that happened to the tower: the lightning that struck the tower, the fire that burned in the tower, and the winds and rain that put it out.
Ho-Oh & Lugia:
The two are thought to be as closely related as they are polar opposites, being based on the eastern legends of the Fènghuáng and Ryūjin.
Celebi: forest spirit. Also able to travel through time, but not the embodiment of time itself
Golem/ Regi trio: Ice, Steel & Rock. Golems in general mythology are creatures that move & talk but are made from minerals. The Regi trio are representative of the elements that make up the earth.
Eon duo: nothing in particular; Lati@s don't have that much of a myth around them. If you really want something, they represent the bonds of love & friendship, considering how much Lati@s care about each other.
Weather trio: ... the weather. Rain, sun & calm (from Air Lock) (Kyogre, Groundon & Rayquaza)
Jirachi: wish Pokemon.
Deoxys: alien life.
Lake guardians: Emotion, Knowledge & Will Power (Mesprit, Uxie & Azelf)
Creation trio: Time, Space & Anti-matter (Dialga, Palkia & Giratina)
Lunar duo: dreams & nightmares (Cresselia & Darkrai)
Manaphy & Phione: sea guardians.
is the embodiment of the planet's boiling core
Regigigas: leader of the Golem Trio and embodiment of the earth itself.
Shaymin: purity, and also the gratitude that Pokemon (& humans I guess) pay towards nature.
Arceus: the beginning (in the Pokemon universe)
Victini: Victini embodies pure energy; it can generate an infinite amount of it that it can share with others through physical contact. Victini is also known as the Victory Pokemon, as its energy can bring its trainer victory.
Swords of Justice: ... justice. And righteousness, I guess.
Kami trio: wind, thunder & fertility (Tornadus, Thundurus & Landorus)
Tao trio: philosophical embodiments of Ying, Yang & Wuji (Zekrom, Reshiram & Kyurem)
Gensect: nothing in particular. Its back story is that it was modified from a fossil by Team Plasma, but it doesn't have the same temperament that Mew2 showed after being influenced by Mew & Ash.
Xerneas: rebirth (edited)
Yveltal: death/ destruction