Arigato "Thank you." See Everything2.
Baka is an insult which can range in severity, but basically means "idiot." See Wikipedia, TV Tropes, or Everything2.
Banzai Literally, "ten thousand years," from the Chinese word wansui. Once used exclusively to honor emperors ("may you live ten thousand years!"), the word became a battle cry in WWII for Japanese soldiers and kamikaze pilots. Today it is used to mean "congratulations," "long life," or "cheers," usually shouted three times as a toast. See Wikipedia or Everything2.
Choudo ii "Just right."
Domo Arigato "Thank you very much." See Everything2.
Douzo "Please do"; "go ahead".
Dozo yoroshiku "Pleased to meet you," or literally, "Please favor me." A greeting when meeting someone for the first time. Or perhaps other times, I dunno, Japanese is confusing. To be formal, add "onegai shimasu." See Everything2 (or this other Everything2 page).
Ganbatte "Go for it!"; "Persist"; "Do your best".
Ganbei "Empty cup." A Chinese drinking toast. See Everything2.
Gomen nasai "I'm sorry." This is less formal than Sumimasen, and should not be used when speaking to superiors. Even more casually, with close friends, one might just say "gomen."
Gozaimasu A word added to a phrase to be polite, when talking to someone who isn't family or a close friend (usually a social superior). For example, "arigato gozaimasu."
Hajime-mashite "How do you do?" or literally, "It's the first time we meet." So definitely I guess this is for the first time you meet. See Everything2.
Hai "Yes." Or more accurately, "That is right." See Everything2.
Hisashiburi "Long time no see."
Iie "No." Or more accurately, "That is wrong." See Everything2.
Kagiya! A traditional exclamation when fireworks are set off. Kagiya was a pyrotechnician who helped popularize fireworks in the early 1800s. See also Tamaya!
Kanpai! "Empty cup." Japanese drinking toast. (Related to Chinese toast ganbei.)
Konbanwa "Good evening." I read that it should be spelled "konbanha," but since I'm already going with the "w" spelling of konnichiwa, I might as well do so for this one, as well. I dunno why I prefer these words this way...
Konnichiwa "Good day." I've read that that spelling is incorrect, that it should be "konnichiha," though I prefer this spelling. And of course transliteration makes it hard to know what's right, anyway. See Everything2.
Maamaa "So-so." Or "Now, now." Or "My, my."
Moshi moshi "Hello" (used to answer the phone). See Wikipedia or Everything2.
Namaste A Hindi greeting (see religion & philosophy page). This is something you hear now and then on Lost."
Nanika atta? "What's up?"
Ogenki desu "How are you?" or literally "Are you well?" Something you should probably only say to someone you know well (in which case you might also just say "genki," see general words page). See Everything2.
Ohayou "Good morning."
Ojama shimasu "I'm intruding." Or "excuse me for disturbing you." A polite greeting when entering someone's home.
Okage-sama de "I'm fine, thanks."
Omedetou means "congratulations." It is used on auspicious occasions, such as birthdays, holidays, graduations, weddings, etc. For example, I used it in my Children's Day banner: "Kodomo no hi omedetou!" One might add gozaimasu to the end of the phrase, to be formal. In the 2003 Christmas episode of Justice League, a Japanese toy manufacturer said to the Flash, "Kurisumasu omedetou gozaimasu." (See festivals page.)
Onamae wa "What's your name?"
Onegai "Please?" You might add "shimasu" to the end of the word to be more formal. If you watch the anime Hikaru no Go, people always say "onegai shimasu" before starting a game of Go.
Oyasumi "Good night." Or "oyasumi nasai." See Everything2.
Sayonara "Good-bye." But not used as commonly as you might think. See Everything2.
Shikata ga nai (or shouganai) "It is inevitable" or "it can't be helped" or "nothing can be done." Actually I wonder if I shouldn't put this on the philosophy page, as it's more a way of looking at life. I think, though, that it's not so much a sense of fatalism as it is a way of accepting (or dismissing) life's little annoyances. Or something. See Wikipedia.
Shitsurei "Excuse me"; "goodbye". Or more politely, shitsurei shimishitsa. See Everything2.
Sumanai "I'm sorry." Informal.
Sumimasen "Excuse me"; "I'm sorry"; "thank you". This could be used to politely get one's attention. It may also be used to apologize for making a mistake or inconveniencing something. Or it can be used to express gratitude. See Everything2.
Tamaya! A traditional exclamation when fireworks are set off. Tamaya was a pyrotechnician (apprentice of Kagiya) who helped popularize fireworks in the early 1800s (and later accidentally burned down a good part of Tokyo). -Anime Insider.
Watashi namae wa "My name is..."
Yatta! An exclamation meaning "I (or we) did it!" Though I guess it could mean other things, depending on the circumstances. Maybe. Anyway, Hiro from the show Heroes is famous for saying it. And there's a music video called "Yatta!" which is somewhat famous on the internet. See Everything2 or Wikipedia.