Depends what you're having the laparoscopy for.
You wear a surgical gown which covers you entirely until you are asleep.
Once asleep, the gown is pulled up, normally to the costal margin (that's the lower border of the ribcage in men, just under the boobs in women). That's fairly standard for general or gynae laps. The top drape goes about this level and covers your boobs and shoulders. Some places peg the drape up to drip stands so your face is uncovered, some places just let the drape drop over your face. Depends on how long the op is with me (for long procedures I peg up, shorter ones I don't tend to).
If you're having a general surgical laparoscopy (e.g. for a gall bladder removal) then the lower drape normally goes across just under your tummy button, covering your legs / hips completely.
If you're having a gynae laparoscopy (e.g. lap and dye test, or diagnostic lap looking for endometriosis) then your legs would be put into low level stirrups and they are covered with separate leg drapes - but your perineal area is exposed because they need to put instruments into the uterus to move it around.
Side drapes go unsurprisingly down each side.
Some places have expensive drapes which are basically a large sheet with a hole in the middle (obviously not where I work!)
You will be covered up again before you wake up. They try to cover as much as possible during the op as you get too cold otherwise.