The lighting supervisor is basically like the TD of the lighting world. In many theatres now it is becoming common to have a lighting supervisor and master electrician as well as a TD and master carpenter. The Lighting supervisor will handle the organization and workability of the lighting design and hand off everything the ME needs to hang and program the show.
The Lighting supervisor is usually the person in charge of managing all of the lighting needs for the theatre. This may include hiring overhire labor as needed, managing the budget, buying and servicing equipment, as well as helping the ME as needed.
The lighting supervisor is often the person who handles all the preliminary work that has to be done with a light plot before it can be sent out on the floor with the ME to be hung. This may include figuring out how each light gets circuited and where data drops go as well as what can get two-fered. They are usually the person responsible for ordering any consumables that are needed like color and templates. They may also be responsible for any rentals that are needed.
In theatres that rent equipment out the Lighting supervisor may be the person who handles that. They are usually the person in charge of the inventory and making sure that they have the equipment needed for their in house productions and working with renters who may need other equipment.