I don't know what the birds represent. I suppose Zapdos represents thunder, so that would probably mean Articuno represents hail and Moltres represents sunshine.
Suicune represents the nimbus cloud, and indirectly rain, Raikou the cumulonimbus cloud, and indirectly lightning, and Entei represents the cirrus cloud, and indirectly sunny weather.
Ho-oh represents the sun, the sky, the rainbow and calm weather in general. He also represents anything symbollically associated with this, such as life and rebirth.
Lugia represents the moon, the oceans and stormy weather. As the polar opposite of Ho-oh, you could also say he also symbollically represents death and destruction. Note that this type of death isn't evil. It is the destruction that creates the room and resources from which life can be created again.
To combine the Birds with Lugia and the Beasts with Ho-oh, I suppose you could say the Articuno, Zapdos and Moltres represent a hailstorm, thunderstorm and a dry storm (which would also explain why they have the Flying type), while Suicune, Raikou and Entei represent a calm downpour, lightning and a calm sunny day respectively.
Groudon represents extreme sunshine, drought and continents.
Kyogre represents extreme downpours and the oceans.
Rayquaza represents perfectly neutral weather and the sky.
The Birds, Beasts, Ho-oh, Lugia and the Titans represent weather. The first and second gen legends as normal Earthly weather, while Groudon and Kyogre are apocalyptic weather, with Rayquaza balancing these two again.
Mew is the ancenstor of all Pokémon.
Mewtwo is supposed to be the most powerful Pokémon, constantly experimented on to become twisted and bend on destruction.
Together you could see these two representing life and death. While Ho-oh and Lugia represent birth, death and rebirth as a cycle, these two represent life and death as the opposite states, sort of like good and evil.
Regice represents the Ice Age, Regirock the Stone Age, while Registeel reprisents the Iron Age.
Regigigas is the force that moved the continents to their current positions. This mostly happened in prehistoric times, so you could relate him the other Regis as the prehistoric period before man existed.
Another interesting thing to note is how rex (-is) is Latin for king or monarch. So they could be seen as the rulers of their respective eras.
Dialga represents the entire time dimension. Not the specific moments, but the entire timeline as a dimension.
Palkia represents the space dimensions.
Together these are obviously the space-time continuum.
Giratina represents anti-matter. You could also say he represents black holes and dark energy.
Arceus represents the Big Bang. The Big Bang is the moment our universe got created, the moment from which time started flowing, space started expanding and energy, matter and anti-matter got created.
Note that Arceus isn't God or the Creator. He is the parent that gave birth to time, space and energy.
Uxie represents knowledge, Mesprit represents emotion and Azelf represents willpower (obviously). They represent the conciousness of humans, although I have no idea how they are supposed to be related to the universe and the Big Bang.
Cresselia represents the full moon and pleasant dreams, while Darkrai represents the new moon and nightmares.
Together these five represent the human nature.
Manaphy represents the sea.
Shaymin represents the fields.
Heatran represents volcanoes.
Together these three represent nature, although I have no idea how Phione and Skymin fit into this.
Celebi represents the flow of time. While Dialga represent the entire timeline as a static dimension, Celebi represents the flow of it, the sequence of moments, both forward and backward.
Jirachi represents fate, possibly also meteorites and falling stars. So you could also say he represents outer space.
Deoxys doesn't represent anything. It isn't a legendary such as the others, merely a mutated virus. If anything, he represents extra-terrestial life, but this doesn't fit in at all with the others.
I have no idea what Latias and Latios are supposed to represent. Possibly feminanity and masculanity.
Also, these are the out of game representations. They are not "gods" or anything. This is just what the creators most likely had in mind when they created the legendaries, to make them seem more special than the normal Pokémon. In the Pokémon universe they are simply animals. More powerful than normal Pokémon, but nowhere near deities.