Let's start with greetings!
Note from Cheri-Sama: These sentences/phrases are commonly heard in subtitled anime series. To those who know or want to clarify something, please tell me. I'll share here how I was taught by my sensei.
Ohayou (gozaimasu) : Good Morning
Usage: Greetings for morning (obviously). If you add gozaimasu it becomes a
polite greeting.
Konnichiwa : Hello / Good Afternoon
Usage: Hello or good afternoon. I remembered sensei told us that they use this greeting for afternoon starting at 10am,
but 10am isn't really afternoon I don't know why though. ^_^x!
Konbanwa : Good evening
Oyasumi / Oyasuminasai : Good night
Usage: when a person is about to sleep/rest for the night. Oyasuminasai is the polite form of Oyasumi.
Sayounara : Goodbye
Doumo (or Domo)/ Arigatou/ (doumo) arigatou (gozaimasu) : Thanks / Thank you / Thank you very much
Usage: Doumo is plain form, the same with arigatou. The polite form is doumo arigatou gozaimasu.
Douitashimashite : You're welcome
Sumimasen : I'm sorry / Excuse me / Thank you (some do use this as thank you)
Shitsureishimasu : Excuse me
Ojamashimasu : sorry for intruding
Usage: I commonly hear this when you go to a friend's/someone's house unplanned (meaning the person
you're about to visit doesn't know you're coming). So upon entering the house, you greet the people there
saying 'ojamashimasu'. This is a polite form.
Gomennasai : I'm sorry
(O)genki desu ka? / genki ka? : How are you?
Usage: Adding 'O' before genki becomes polite.
Dou shimashita ka? : What's the matter?
Odaijini / Ki o tsukete : Take care of yourself.
Usage: (needs clarification again) I think Odaijini is the polite form but kiotsukete is commonly used.
Okaeri / Okaerinasai : welcome back
usage: Used to welcome back (commonly at home). Polite form is okaerinasai.
Tadaima : I'm back
usage: Used by the person who just came back (from work, school, etc.)
firefly addendum:
Ittekimasu: I'm going
Usage: Said when a person is leaving to go somewhere, but with the intention that they will return.
Itterasshai: Have a good time; See you later
Usage: Said to people when sending them off from the home or some other place where they will be returning at a later
time
Hajimemashite: How do you do; Nice to meet you
Usage: Used when introducing oneself.
Yoroshiku Onegaishimasu: Please treat me kindly
Usage: Most commonly used at the conclusion of an introduction.