The upcoming Pacific Rim is probably the first real dose of something the Japanese call kaiju that the US has gotten in a long while. The word means "strange beast" in Japanese and generally refers to what we call monster movies. However in Japan they have essentially perfected the film genre, removing it from the niche or supplementary role of some films, and pushed it into the stratosphere of our imaginations. With the film coming out so soon it couldn't hurt to understand the genre more.
Godzilla, or Gojira in Japan, was the first major kaiju film and set a standard that still has yet to be broken. Not only would Godzilla start a trend of more and more monster movies he would usually be the monster in them. Eventually the popularity of Godzilla saw him defending Japan and Earth more often than destroying it. The original film was a heavy dose of reality for the Japanese people who create this monster, itself a product of radiation, less than a decade from being attacked with two atomic bombs. When the film was initially released it got some negative reviews for this focus on radiation, some saying it was exploitative of the Japanese mindset so close after WWII and other radiation exposures since the war. The film quickly outpaced it's opening critique and went on to become the genre defining film we know it as and spawned 27, and counting, sequels.
While the original is a classic and should be watched at least for cultural purposes, but it isn't readily available for everyone. Luckily you can catch some of the decent sequels on your local streaming services like Netflix and Amazon. Over on Netflix you can watch Ghidorah: The Three Headed Monster and Godzilla vs. Monster Zero (also on Amazon). Both are within the initial Showa period of Godzilla films, which means they were produced before Hirohito died, and begin the transition from terrible monster to anti-hero for Godzilla. Also on Amazon you can find Godzilla: King of All Monsters, an American production released a few years after the original and re-dubbed the Japanese film to introduce the American audience to this new beast.
America has tried it's best to copy the kaiju style and in some cases were successful. More recent examples include J.J. Abram's Cloverfield, which just has a giant, mysterious, monster tearing New York apart. You're treated to the experience through the lens of a camera as a small group of friends rush around the city looking for one of their own. The film tries to build tension and mystery as it doesn't reveal the monster until late into the movie but a lot of the monster movie tropes are there. Some alternative American choices can also be found on streaming.
Super 8 (on both Netflix and Amazon) is a Spielberg and Abrams joint effort which again combines mystery with an alien monster. It isn't a perfect film but revives a feeling in movies, following a group of kids around on an adventure, that only Spielberg himself seemingly mastered. Monsters (only on Netflix) follows two people as they try to travel out of quarantined Mexico which has become infested with alien monsters. It's more a tale about the two people trying to escape rather than the monsters themselves but is still a fun watch.
There are loads more films about monsters in the same kaiju vein and feel free to share your favorites in the comments. These were just the few I think you should catch before heading out to see Pacific Rim this weekend.