Tere: this is equivalent to "hello". Informal ways to say this are "tere-tere" and "terekest." Etymologically, "tere" is related to the word "terve" (healthy, whole), which was used as a greeting as far back as the 12th century.
Tervist: another way to say "hello", again related to the word "terve" (healthy, whole).
Tere hommikust: equivalent to "good morning". The complete phrase, which is no longer used in Estonian, was originally "terve hommikust aega" (good or healthy morning time), a greeting which goes back to the twelfth century or earlier. Modern variations of the morning greeting which can also be used in spoken language include "tere hommikut," "hommikust," "hommikut," and "hommik".
Tere päevast: equivalent to "good day". This is the greeting normally used in the afternoon, between morning (tere hommikust) and evening (tere õhtust). Informally, you can also simply say "päevast".
Tere pärastlõunat: this means "good afternoon." It is not as common in Estonian as "good afternoon" is in English, though. Other variations are "tere pärastlõunast" and "tere pealelõunat," and there is also "tere lõunast" and "tere lõunast" for noontime or midday, although these are not common greetings either. It's much more common to say "tere päevast" (good day) at any time between morning and evening.
Tere õhtust: equivalent to "good evening". Like "tere hommikust", this greeting originally meant "good/healthy evening time," and in the 12th century was probably something like "terveh õhtagosta aigaa;" by the 15th century it had evolved to "terve õhtust." Variations used in modern spoken language include the shorter versions "õhtust" and "õhtut".
Tere kõigile: this means "hello everyone", so you can use it to speak to a group. You can also add "kõigile" to any of the other greetings above: "tere hommikust kõigile" (good morning everyone), "tere päevast kõigile" (good afternoon everyone), etc.
Tere tulemast: this means "welcome." If you want to say "welcome to someplace", use the illative (sisseütlev) case: "Tere tulemast Eestisse" (welcome to Estonia), "Tere tulemast minu blogisse" (welcome to my blog), "tere tulemast koju" (welcome home). "Welcome back" is "tere tulemast tagasi."
Jõudu tööle: this is a greeting that can be said to a person who is working, and it means "strength to the work." Variations are "tere jõudu" (greetings of strength) "jõudu teile" (strength to you) or simply "jõudu" (strength). The reply is "jõudu tarvis," which means "strength is needed." Another way to answer is simply "tarvis" (it's needed).
Jätku leiba: this is a greeting that can be said to a person who is eating. It means "may the bread continue" or "may the bread not run out." Another way to say it is "jätku leivale." "Jätku toitu" and "jätku toidule" (may the food not run out) are less common. The reply to any of these can be "jätku tarvis" (it needs to continue; the continuation is needed) or simply "tarvis" (it's needed).
Tšau: this is an informal greeting and it can also be used to say "goodbye." Other ways to spell (and pronounce) it include "tšao," "šau," and "tsau." It comes from the Italian word "ciao".
Hei: this can also be used an informal greeting. It's a loan-word from Swedish, and it's more commonly-used in Swedish than it is in Estonian. A variation is "hei-hei", and an even more informal version is "heippa".
Kuidas läheb?: this is equivalent to "how are you?", and literally it means "how does it go?" or "how's it going?" To say "how's it going for you?", just add the pronoun: kuidas sul läheb?, kuidas sinul läheb?, kuidas teil läheb?
Mul läheb hästi: this works as an answer to "Kuidas läheb," and means "it goes well for me" or "everything is going fine." You can also say "Minul läheb hästi" (it's going fine), "Minul läheb hästi, aitäh" (it's going fine, thanks) or simply "Läheb" (it's going).
Aga sul?: this means "and you?" and can be used with "kuidas läheb?" or related questions that use "läheb," "läks," and "läinud." The formal version is "Aga teil?" There is also the longer version, "Kuidas sul endal läheb?" and "Kuidas teil endal läheb?" (and how's it going for yourself?)
Kuidas käsi käib?: this is another way to ask "how are you?" Since "käsi" means both "hand" and "arm" in Estonian, this can be literally translated as "how does your hand go?" or "how does your arm go?" (There is also a plural form, "kuidas käbarad käivad?", which means "how do your hands/arms go?") This is an older greeting, but it is still used.
Kuidas sa elad?: this is an informal way to ask "how are you?", meaning "how do you live?" You can also say "kuidas elad?" or, more formally, "kuidas teie elate?" or "kuidas elate?" More informal ways to ask this are "kuidas elu läheb?", "kuidas elu veereb?" or "kuidas eluke veereb?" (how's life going?)
Hästi: this is a good answer to any of the "how are you" questions (kuidas läheb, kuidas käsi käib, kuidas sa elad, etc.), and means "fine" or "well". You can also say "Tänan hästi" (fine, thank you) or "Tänan küsimast, hästi" (fine, thanks for asking). If you are doing very well, you can say "väga hästi" (very well) or "päris hästi" (quite well). "Kõik on korras" means "everything is fine" or "everything is alright." Other answers include "normaalselt" (okay), "pole viga" (not too bad) and "keskmiselt" or "keskeltläbi" (average; so-so).
Aga sina?: this means "and you?" and can be used with "kuidas sa elad?" With the more formal "kuidas teie elate?", use "aga teil?"
Mis uudist?: this means "what's new?" You can also say "mis teil uudist?" or "mis sul uudist?" (what's new with you). A possible reply is "mitte midagi" (nothing). Other replies include "mitte midagi erilist" (nothing special), "ei midagi erilist" (nothing special), "ei midagi" (nothing), or "ei midagi, tänan küsimast" (nothing, thanks for asking). "Mis toimub?" means "what's happening?".
More greetings (and a source for some of this information): Greetings in Languages of Estonia.