SPOILERS! SPOILERS! SPOILERS!
Baring any explanation I missed since I do not understand a great deal of Japanese yet, here is what I know about Godzilla's origins from seeing the film twice:
In 1954 (or at least around that time), a bizarre life form was discovered feeding on nuclear waste dumped into the sea. Japanese scientist, Goro Maki, names it Gojira after a mythical monster god revered and feared on his home island of Odo. He ultimately teams up with the American government, who are highly interested in studying the beast, which they code name "Godzilla". Time passes, and through a mysterious and still unknown sequence of events (maybe we will learn more from the sequel?), Goro Maki commits suicide circa 2016, leaving his few belongings and data on Godzilla on his boat, which he leaves floating empty in Tokyo Bay. Soon after the boat is discovered, an eruption of a mysterious red liquid rocks the bay, and soon after a tail emerges from the water. Like a slow moving tsunami, the initial form of Godzilla makes way inland, ultimately evolving into a second form before escaping back into the sea, evolving again, and returning to destroy Tokyo in his final form.
The exact origins of the beast, and what happened between his initial discovery and his emergence in Japan in 2016, are (as far as I can tell) left a mystery in the movie. It is known that the monster's cells can reproduce asexually, like Sanda and Gaira's cells from War of the Gargantuas, and that, at the end of the film, smaller monsters seem to be emerging from his tail.
We may have to wait for a sequel to get more answers. :)