Looking after your drunk friends when you are the only sober one. Keeping an eye on them, escorting them everywhere, keeping them out of trouble and sometimes cleaning them up. This should also include making sure they get home safe with all the belongings they brought with them.
"Yeah, the dance was fun but I spent pretty much the whole night drunk-sitting Mary! She wouldn't stop telling people how drunk she was."
The act of looking after a friend who after a long night of drinking has regressed back to an approximate mental age of five. This can include taking away their phone to keep them from making calls/texts they will regret, keeping them from going to more parties/drinking more, giving them food to eat/feeding them, making sure they don't choke on their own vomit if they throw up.
Tanner: *has been drinking vodka all night* I'm going to another mixer!
Anne: Like hell you are!
Cami: You need to go to your room and sober up.
*Anne and Cami then take Tanners shoes and cellphone to keep him from going to the party, and bribe him with Moo Shu and Metalocalypse
to ensure that he stays in for the night. *
the above is a perfect example of drunksitting