A person in my party punched a bartender and I didn't know how to calculate how much damage it would deal. Does anyone know what die I use to calculate this and if any modifiers increase the damage?
1 + Str modifier, as the SRD says
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head--butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
Some features (monk, etc) may change the amount of damage caused.
The current PHB errata from WoTC updates what's in the PHB on page 195, and states the unarmed strike proficiency as shown in the SRD. It also states:
Weapons (p. 149). Unarmed strike doesn’t belong on the Weapons table