> hi! my name is____ = konnichi wa watashi no namae wa (name)
>whats your name? = anata no namae wa?
>nice to meet you = hajimemashite(as in first time), yoroshiku onegaishimasu(as in becoming acquainted). the person who answered first did a good job explaining how this works
>nice to meet you too. = kochira koso
however if i were to do a normal introduction in japanese it would be like this
person 1>hajimemashite=its nice to meet you
person 2> hajimemashite= its nice to meet you too
person 1>watashi no namae wa (name) =my name is (name)
person 1>anata wa? =whats yours?
person 2>watashi no namae wa (name) = my name is (name)
person 2>kyou kara watashi wa kono kaisha ni hatarakimasu = from today I'm going to be working at this company (just a little conversation about why they're meeting)
person 1> sou datta jya yoroshiku ne= is that so, then lets work well together then (yoroshiku can be translated into many different things in english)
person 2> hai, yoroshiku onegaishimasu. = yes, id also like to work well together.
this is just an example of what a normal introduction might be in japanese but since im assuming your starting out i thought it might be immediately useful for you to see it in context. as well as giving you the translations