It started like kind of funny exercise, when I asked myself, how many ways you have in modern Hebrew to say "hello". Then I recognized it might be a good idea to write a short lesson in this way. (Another interesting question may be, how many ways there are to reply.)

So, let's talk about greetings in Hebrew. Typically, it's one of the first things somebody wants to learn, when trying to approach a new language.

1. Shalom!
 

!שָׁלוֹם עֲלֵיכֶם  [shalom aleikhem] or [shalom aleikhem] - Literally: "Peace upon you!" - this one is the most traditional one, and probably is samiliar to everybody, as well as the next one:

!שלום - [shalom] - Most common greeting; but it is in fact simplification of the former. You can also say "shalom, shalom!" Well, I guess, everybody knows what does this mean.

2. What's up?
 

?מַה נִשְׁמַע  [ma nishma] - "what's up" - is probably second by popularity among the greetings, if not the first.

There is a shortened version of it, it's quite new, and it's rather slang of Tel-Aviv rather than common phrase everywhere:
?מה ניש [ma nish?] I personally think it doesn't sound the best from aestetical point of view... :) But maybe I'm just too old-fashioned...

3. How are you?
 

?מַה שְׁלוֹמְךָ  [ma shlomkha] (m) - "How are you doing?" (when talking to male.)

When talking to female: מַה שְׁלוֹמֵךְ? [ ma shlomekh] (f)

Also, sometimes you can hear variations:

?מַה שְׁלוֹם כְּבוֹדוֹ - [ma shlom kvodo] - "How Your Honor is doing?" - either very respectful or a little ironic.

?מַה שְׁלוֹם הַגְּבֶרֶת - [ma shlom ha-g_everet] - "How are you doing, Ma'am?"

?מַה שְׁלוֹם גְּבִרְתִּי - [ma shlom gvirti] - "How are you doing, (my) Lady?"

?מַה שְׁלוֹם הָאָדוֹן? - [ma shlom ha-adon] - "How are you doing, Sir?"

?מַה שְׁלוֹם אֲדוֹנִי - [ma shlom adoni] - "How are you doing, (my) Sir?"

4. Good morning!
 

בּוֹקֶר טוֹב - [ boker tov ] - Good morning! - and we answer:

!בּוֹקֵר אוֹר - [ boker or ] - "Morning (full) of light!"

עֶרֶב טוֹב  - [ `erev tov ] - Good evening!

Variations (again, all those are greetings, not farewell):

!בּוֹקֶר מְצוּיָּן - [ boker metzuyan ] - (Have an) excellent morning!

!עֶרֶב נָעִים - [ `erev na`im ] - (Have a) nice evening!

Remark: נָעִים is not plural; the plural of נָעִים would be נְעִימִים

!עֶרֶב מְצוּיָּן - [ `erev metzuyan ] - Excellent evening!

צָהֳרַיִם טוֹבִים - [ tzohorayim tovim ] - Good afternoon (literally: Good noon, since צָהֳרַיִם means "middle of the day") Since צָהֳרַיִם is plural (rather dual), the adjective טוב gets its plural form טובים.

If you're trying to be polite, don't forget to add something line "Sir", or "my Lady", as mentioned above.

Well, although it's not exactly greeting, but for "Sweet dreams" we say in Hebrew:

לַיְלָה טוֹב  - [erev tov] - Good night! (same as Sweet dreams!)

You also can say שֵׁינָה טוֹבָה - [ sheina tova ] - literally: (Have a) good sleep!

What about saying "have a good night" as farewell? Let's keep this as possible topic for the next article.

5. What's going on?
 

?מַה קוֹרֶה - [ma kore] - "what's going on"

You may also say "ma kore itkha" (m) / "ma kore itakh" - literally it's "what's going on with you?" - but this phrase in English might sound rather like "what's wrong with you?" The Hebrew phrase "ma kore itkha?" may be use both as greeting or (with slightly different voice, huh) as "what's wrong" phrase too.

אִתְּךָ - [ itkha ] - "what's going on"

אִתָּךְ - [ itakh ] - "what's going on"

?נוּ, אַז מה קורה איתך - [az ma kore itkha, habibi] - "So, what's going on (with you)?"

6. How's everything?
 

 ?מָה הַמַּצָּב - [ ma ha-matzav? ]  - it's translated as "How's eveything?", or literally "what's the state (of the affairs)?", while the word מצב means literally "situation, state".

7. Yet Another How's everything?
 

 ?מָה הָעִנְיָנִים - [ ma ha-inyanim? ], and with "ha-" part disappearing, it turns into

?מה עניינים - [ ma `inyanim? ]

The most adequate translation of עניינים in this context is probably "all kinds of things". "How's everything?", that's what the phrase means.

In general, עניינים is plural of the word עִנְיָן [ `inyan ]  which has the meaning of "matter, business", and also "interest (in something)". (The letter Yud might be doubled when writing without vowels.)

?מה עניינים, אחי -- How's everything, (my) brother?

8. Welcome!
 

!בָּרוּך הַבָּא - [ barukh ha-ba ] - it means Welcome!, and literally translates as "Blessed is the comer". Somebody is coming in, and those meeting him (in their house, for example) are blessing him, that's what it is.

!בְּרוּכִים הַבָּאִים - [ brukhim ha-ba'im ] - Same thing as the former, but the greeting is being told to more than one person.

!בְּרוּכִים הַנִּמְצָאִים -[ brukhim ha-nimtza'im ] - Although rarely used nowadays, this would be a really nice reply to those who are telling you Welcome. Literally it means "Blessed (are those, who are) residing (here)!"

!בָּרוּך הַשָּׁב - [ barukh ha-shav ] - Another idiom of the same pattern; this is how you welcome somebody who had gone and then returned: "Blessed (is) the (one who is) returning!"

9. How's going?
 

?אֵיך הוֹלֵך - [ eikh holekh ] - Yep, it's almost literal translation of either English "How's going?" or German "Wie geht es?" (which means exactly the same.)

10. What's new?
 

All the variations of "anything new?", when you expect some news:

?מַה חָדָשׁ - [ma chadash?] - "What's new?"

?נוּ,יֵש חֲדָשוֹת - [ nu, yesh chadashot? ] - "Any news?" The "Nu" is optional, of course. :)

Some Extras:

You can say just: Hi!

Do I really need to translate it? :)
Let me just bring the Hebrew spelling of this word:

!הַי

Obviously, this one is borrowed from English. And another one, quite popular, was borrowed from Arabic:

אַהֲלָן - [ ah_alan ] - hello - is a short version of Arabic "Ah_alan wa-sah_alan" (Welcome).

As you can see, even these "10 ways" are giving you enough of learning materials. :) Enjoy!