Tag is a children's game. Of a group, one person is said to be..."IT". It is the IT player's duty to chase after the others and find one other person to touch or "Tag". Then, the newly touched person is IT, and they must chase down another person and tag them, and make the new person IT. It is rather fun for children.
The phrase, "Playing Tag" carries over into adult life, when it is used to refer to adults to continually try to make contact in person, or my phone or other messaging. If they seem to be unable to make contact, they refer to their situation as "Playing Tag".