Many of the names you might encounter in Persian are Arabic in origin and/or by convention, and it pays to be familiar with the structure of Arabic names in order to understand Persian names. Strictly speaking, however, Persian names are quite similar to English names in that they have two basic parts, the given name (نام, nām, or the Arabic اسم, ism) and (since the early 20th century) the surname (لَقَب, laqab, from Arabic, or نامِ خانِوادگی, nām-i khānivādigī, “family name”). One cultural difference that you encounter in Persian names as compared to Arabic names is that, while lots of Persian-speakers have very Arabic-sounding first names, many other (more culturally “Persian,” I guess you could say) first names are taken from the Shahnameh.
The typical Iranian surname is something like the Arabic nisbah or laqab in that it describes something about the person (or one of the person’s ancestors, whoever “founded” the family line): maybe a profession, a personal quality, or the city/region/district from which the person or family hails. These may take one of several suffixes; the most common are ی (-ī), which designates an ancestral or geographic marker, پور (-pūr), meaning “son of [the name to which it’s appended],” زاده (-zādah), meaning “descendant of…,” and نژاد (-nizhād), meaning “from the tribe or clan of….” Our friend Mahmoud Ahmadinejad’s last name (احمدینژاد), for example, means “from the clan of Ahmad,” “Ahmad” being another name for “Muhammad” (both derive from the same Arabic root).
Also, in terms of asking for a name and giving yours, “What is your name?” is simply نامِ شُما چیست؟ (nām-i shumā chīst), and in response you’d either say “I’m _____,” _____ مَن (man _____), or “my name is _____,” نامِ مَن _____ است (nām-i man _____ ast).