Within each topic, the phrases are listed from least to most polite, with the standard form(s) in bold. If there are more than two or three forms listed, it's not really clear which is the most polite.

The phrase... Means... Literally means...
Oyasumi nasai
"Good night." Say this to someone when they're going to bed, or are likely to go to bed soon. If you say it to the lads at the bar when you hang it for home, you're implying that they're going to sleep there. That may be true, but it's not polite to point it out; use "***" instead. Please rest.
Ohayou gozaimasu
"Good morning." It's early.
Arigatou gozaimasu

Arigatou gozaimashita
"Thank you." *** When someone does something for you, you say, "Arigatou gozaimasu." ("You are going to a lot of trouble" or "Thank you for what you're doing right now.") When you meet them a little while later, you say, "Arigatou gozaimashita." ("You went to a lot of trouble" or "Thank you for what you did.") Polite people say "Arigatou gozaimashita" even if they already said "Arigatou gozaimasu." There's no past tense of plain arigatou, but polite people do say "Arigatou" after the fact as well as at the time that someone does something for them. You are going/went to a lot of trouble.
Dou itashimashite "You're welcome." "Not at all."  
Gomen nasai
"I'm sorry." This can sound rather childish; children use it all the time, but adults say it mainly to people who are above them.  
"I'm sorry." This is more adult than gomen nasai.  
Itte kimasu
Itte mairimasu
"I'll be back." Say this only when you know you'll be back--when you leave home, work, the bar, the tattoo parlor, whatever. If you're leaving the dentist, say "***". I'm going out and coming back.
Itte irasshai "Be back soon." The response to itte kimasu Go out and come back.  
Tadaima "I'm home!" The Japanese say this every time they come home. Just now
Okaeri nasai
"Welcome back." The response to tadaima.  
Ojama shimasu
Ojama itashimasu
What you say when you enter someone else's home.

For a cool scene which turns on the use of the last few phrases, watch the scene in Evangelion where Misato brings Shinji home for the first time.

I'm going to bother you.
Ojama shimashita
Ojama itashimashita
What you say when you leave someone's home. I bothered you.
Itadakimasu What you say before you eat. Translators go through contortions to translate this: "Let's eat!" "Dig in!" "It looks delicious!" "Shall we begin?" I am about to partake.
Gochisou-sama deshita What you say when you're done eating. Rare in anime, perhaps because anime characters eat like Barbary savages and perhaps because they tend to blow everything up before the soup course.  
Omedetou gozaimasu
Shou ga nai
Shikata ga nai
"There's nothing to be done for it." There's a strong feeling of fate in this sentence, along with the resignation: "That's the way things are, and I can't change them." Some people say that this phrase encapsulates the Japanese philosophy of life.  
Konbanwa "Good evening." You don't use this with members of your in-group--your family, classmates, workmates, party members, whatever.  
Konnichiwa "Good day." You don't use this with members of your in-group--your family, classmates, workmates, party members, whatever.  
Saraba "Good bye"  
Ja or Jaa "Bye." Ja ne = "Bye." Ja mata = "See you later."  
Baibai "Bye-bye." This doesn't sound nearly so silly in Japanese. After all, it's English, which is the coolest, hottest, sexiest, and most thrilling language around--so it has to be chic. Not used on formal occasions, though, as you might have guessed. English - bye-bye
Sankyu "Thank you." English - thank you
Hajimemashite "Pleased to meet you." [I am meeting you] for the first time.
Douzo yoroshiku onegai shimasu
"Pleased to meet you." Please treat me kindly.
Moshimoshi "Hello?" (when answering the phone)  
Onegai shimasu "Please" Please grant my wish.