The term "tripping the well" got it's start most likely as a
reference to the "round trip" made by the drill bit. The first leg
of the trip is coming out of the hole, by lifting the entire drill
string and unscrewing the upper most drill pipe called a joint, the
newly removed drill pipe is laid beside the rig and the drill
string is lifted again, etc. until all drill pipe is out of the
hole. After service work is performed, for example change to fresh
drill bit, then the second leg of the trip is to reverse the
process, lowering the drill string as each new joint is made until
fully back in the hole with the bit on bottom. You can find a lot
of interesting videos on offshore rigs, jobs, etc here:
I left the previous answer in place:
When drill pipe is lowered or put into the well in the well, it
is called running in or is tripping into the well. When drill pipe
is raised or pull out of the well, it is call running out or
tripping out of the well.