Sure, there's plenty of resources out there (like the Wikipedia link above) to explain Kaiju fandom terms, but I figured it might be polite to include a little something here. Plus, I may not always use terms the same way that most do, so consider this a "Dave to The Rest Of The Fandom" translation guide. It's short, so the ordering is more of a "more important to less important" sort of thing, rather than alphabetical.

• KAIJU - Giant movie monsters. Traditionally guys in rubber suits, but lately they're more likely to be CG creations. Godzilla is the ur-Kaiju, the King of Monsters. Also called Daikaiju, which is more correctly giant monsters.
• TOHO - A Japanese studio responsible for most of the Kaiju out there. They did Godzilla an all of his buddies, plus semi-related movies like the Mysterians, War of the Gargantua, etc. You may sometimes see me talk about the Tohoverse, a sort of loose agglomeration of all their monsters. A lot of movies in the Tohoverse are connected in some way to Godzilla without ever featuring him.
• DAIEI - While Toho is the undisputed number one in kaiju, Daiei was probably their closest competition, with their giant turtle Gamera. Daiei went defunct a while back, and the Gamera rights were sold to Kadokawa. Daiei wasn't nearly as prolific as Toho, and as far as I know there were never any Gamera-verse movies without Gamera in them.
• KAIJU AGE - Or, as it's often spelled in clumsily romanized form, "Kaiju Eiga". Just as comics are divided into Golden Age, Silver Age and so forth, Kaiju movies are split into (so far) three ages.
  1. SHOWA - Generally seen as running from the original Gojira/Godzilla in 1954 through about 1975. The last Showa movie was Terror of Mechagodzilla, but Showa-style kaiju would continue to appear on TV (i.e. Ultraman) and comics (Marvel's Godzilla comic) for several years to come. One could consider Gamera, Super Monster to be the last of the Showa movies, but it's really just a clip show and most fans who even know about it would rather forget it. Showa kaiju are either guys in suits ("suitimation") or puppets on wires, and the effects are sometimes deliberately cheesy. Early Showa was all about horror aspects, and kaiju as metaphors for atomic devastation. Later Showa tended to have kid-friendly monsters acting as protectors...Gamera made the monster-to-friend transition very quickly, while Godzilla was still sort of an uneasy ally through most of the age.
  2. HEISEI - 1984-1994, give or take. The starting mark was the kaiju revival of Godzilla 1984 (1985 in America), and the ending mark is generally seen as Godzilla vs. Destroyah in 1994, although the American Godzilla (GINO, or Godzilla In Name Only) is a bit of an odd duck that fits in neither Heisei nor Millennium. Gamera was also revived for three movies in this age, which was marked by vastly improved rubber suit technology and a return to darker storylines. Godzilla and Gamera were still generally defenders, but they were far less concerned with actually protecting individual people.
  3. MILLENNIUM - Kicking off with Godzilla 2000, and sometimes called the Shinsei Age. This is the age we're currently in, and it's been marked by movies where only the orginal Gojira (or the original Gammera The Invincible) is kept as backstory, so lots of one-off takes (although Godzilla X Mechagodzilla and Tokyo S.O.S. form a two-parter). A lot of experimentation is done in this age, such as making Godzilla the bad guy and Ghidorah the hero, or replacing Gamera with his son.
• K-GIRL - An anthropomorphic female representation of a kaiju, generally with heavy sexual overtones. Taken from Twisted Kaiju Theater. This site is not particularly safe for work. Or sanity in general.
• NAME ISSUES - Sometimes a monster goes by multiple names, and there's some monsters with rather similar names. There's a lot more than this list, but I'll only add someone if I talk about them under one name or another.
  • GOJIRA/GODZILLA/GINO - Gojira (a sort of gorilla-whale thing) is the original name, Godzilla is the Americanized name. Same guy. GINO, or Godzilla In Name Only, is what fans took to calling the 1998 American Godzilla, although Toho has since given it the official moniker of Zilla.
  • GAMMERA/GAMERA - The original movie spelled it with two M's, but the extra M was dropped pretty quickly. Just including this so no one thinks "Gammera" above is a typo. Japanese for turtle is "kame", the likely origin of this name.
  • GAOS/GYAOS - One of Gamera's foes, I think it's mostly a fan construction to spell the Showa version Gaos and the Heisei/Millennium versions Gyaos, but I could be wrong. I like to pluralize it as Gyaosim, but I don't think anyone else does.
  • BARAGON/BARUGON - The first is a Tohoverse monster, the second is a Gamera foe with a battering ram tongue that spits ice and a rainbow beam from his back that can slice stuff. Baragon is merely a monster, Barugon is someone's bad acid trip.