First, you will hear people refer to themselves as بنده bandeh, servant/slave, instead of من man, I. Men or women can use it, but I feel like it's more commonly used by men. (You may have even heard some people use ما mā, we, to mean I. This is not a sign of humbleness. It's fine to use with friends and family jokingly but sounds pretty pompous if you are serious when you say it. The Shah of Iran was known for using mā, we, when he meant man, I.) In this clip from the movie To Va Man, Behnoosh Bakhtiari uses bandeh instead of man when she offers to stay with her friend, saying bandeh khodam mimunam, I'll stay myself.
Once I met someone, and after telling me his name, he followed up with [کوچک شما [هستم, kuchike shomā [hastam], literally, [I am] your small. Clearly in these examples, the person is trying to be humble and acting as your inferior. Sometimes people will even drag their poor children into this mess and say they are your inferior, too. You can respond with something like noore cheshme mā hastin, you are the light of my eyes, or āghāin, you are a gentleman.
When people are proud of you or want to express just how special you are for them, they may say, tāje sare mā hastin, you are the crown on my head. In a way, they are implying that they are inferior to you. You can reply, اختیار دارید ekhtiār dārin, you have the authority to do with me what you will. This, in turn, shows your inferiority to them. See the game?
Another popular word is nokaretam, I am your servant. I've been hearing this a lot on the much-loved TV series Pāytakht. One of the main characters, Naghi, says this to almost everyone he meets. As he says it, he places his hand over his heart, which we know is a gesture of sincerity.
On a bit of an unrelated note, this past weekend, a friend asked me how long I had been blogging, and I said for about a year now. It got me thinking, so I came back and checked my posts only to find out that I missed my own one year anniversary on April 21! (And my second post was The Art of Taarof, Part 1- Way to keep some balance!) So after one solid year, I'd like to say thank you to all of you who read, follow me on Facebook, Instagram, Twitter, or Google+. From the bottom of my heart (and I place my hand over my heart as I say this) Bandeh nokare shomā hastam!