what's the meaning of "(yelling over to bartender): Another round and back it up for me and Lily."

Question

Answer

They are asking for 1 round of drinks but two with one extra each for the speaker and Lily.

So the speaker and Lily want two drinks each and the rest of the party will have one more drink each.

Answer

by the sound of it, the person is asking the person who works behind the bar for the same drinks that they had ordered for themselves and the same drink Lily had