Wie geht's?, "How goes it?", is the typical way of asking, "How are you?" in German. The full form is, Wie geht es dir/ihnen?, "How goes it for you (informal/formal)." The full form is usually used in formal situations, while the dir is often dropped and the es contracted to 's in informal situations.
Two important grammatical notes about this construction must be made. First, this construction is indirect: one should not say, Wie bist du?/Ich bin gut.; these have a different connotation, either "How well behaved are you"/"I am well behaved," or "How are you in bed?"/"I am good in bed." It is not literally asked how one is, but how it goes for one. Second, the personal pronoun here falls in the dative case (mir/dir/ihnen/ihm/ihr/&c.).
Replying to Wie geht's? is quite simple. One can answer merely gut/schlecht, "good/bad", though the long form is Es geht mir gut/schlecht. Another common answer is Es geht so, or Geht so, roughly "It's going okay." One important note here is that if you thank the speaker for their concern, the thank you usually comes before the response, i.e.: "Wie geht's?" "Danke gut, und ihnen?" "Danke gut." Even more odd for English-speakers is the simple response, Danke, dir? I find that this can be most easily parallelled to the, "How you doin'?" "How's it goin?" interaction, in which no information is actually exchanged. Also, don't expect Danke gut. to be the normal response, even when one isn't well; if one isn't doing so hot but doesn't want to discuss it, their response will likely be, Danke, und dir/ihnen?