The Poles are a friendly bunch. They love to meet new people, listen and learn from them, and always enjoy sharing a drink, whether it's with a stranger or a friend! For the likeminded visitor to Poland, being able to introduce yourself confidently is really important, and in Polish there are (as in English) a whole host of different ways to do this. Some are formal, others are relaxed, some will sound out of place in certain situations and some - which unfortunately appear in a very many Polish language textbooks - will just sound horribly anachronistic.
I remember reading my first introduction to Polish phrases, just before I first went to Kraków. Full of confidence and gusto I must have introduced myself to countless people, fellow learners and strangers with an overwhelming, Victorian-esque, bourgeois style, fit only for use in the quaintest Polish circles today; probably more suited to introductions in the medieval aristocracy. "Hello gentleman, my name is James and my surname is Bond;" imagine hearing this rather encumbered introduction from a complete stranger? A bit too much perhaps.
Like many other European languages, Polish has a formal and informal way to address people. If in doubt, it's always better to be over polite, and for this thepan, pani form of address should be used. However, when speaking to friends, family and younger people, you can use the impolite form, using instead the relevant personal pronouns (or none, as they're normally omitted in Polish anyway) to start your sentences and questions.
For example, to formally ask someone their name, you would use the sentence, 'jak sie pan/pani nazywa?', but talking casually or informally you can omit both, the polite pan or pani, and indeed the pronoun 'your', saying simply, 'jak sie nazywasz?'.
The simplest way to introduce yourself in Polish is to use the verb 'to be' - być - in the first person singular: Jestem Asia (I am Asia). This is particularly easy to grasp for English (and most European) speaking learners, because we do precisely the same thing with the English verb 'be', when we say 'I am'.
To introduce someone else you can also use the verb 'to be', conjugating it to match the new subject of the sentence, and, instead of using a pronoun, modifying your starting word to indicate an introduction (adding the pan/pani pronouns for formality if necessary) : To jest moja żona (this is my wife). To ask who someone is, you can use the nominative questioning kto (who): Kto to jest? (who is that).
To ask a first name only, you can use the question, jak masz na imię (literally meaning, 'what first name do you have'). This is answered either by using the first person form of być, as we have seen, or by the full sentence, mam na imię... (literally meaning, 'I have the first name...').
After you've met someone, and if you so wish, you can leave saying bardzo mi miło (very nice to meet you), or bardzo mi miło pan/panią poznać for formal departures.