When you're in some kind of meeting or party, the phrase "go around the room and ___" means for each person to do something, one by one. For example:
Let's go around the room and introduce ourselves. I'll start.
This means that each person in the room is going to introduce himself or herself. The speaker is going to go first, then another person, then another person, and so on.
Here's another example:
I'll go around the room and ask everyone to vote.
This means that the speaker is going to walk to each person in the room, one-by-one, and ask each person to vote for something.