The SPE does a book "Petroleum Exploration in non- technical language" (or something), which would give you a good introduction to the process. Briefly:
Geologist and geohpysicsists study a region and decide where to drill. The studies include the low tech stuff like looking for oil leaking out of the ground to high tech stufflike magnetic surveys, gravity surveys and (mostly) siesmic surveys, where sound waves are bounced off the rock to form an idea of teh layers and structures int eh rock underground.
The you drill a hole (sounds easy doesn't it?) with a drill rig: offshore these can stand on the sea floor (jackups) or float around (semi subs and drill ships). The hole can be vertical or horizontal or anything in between and nowadays complex, 3 dimensional well trajectories are not uncommon. While you're drilling you're looking at the cuttings coming back out of the hole, looking for the type of rock and traces of oil. After you've drilled the well you log it: you lower special tools into it to measure things like the electrical conductivity of the rocks, the radioactivity of the rock and so on, form which you can determine the type of rock, rock density and if there's oil in the rock. Sometimes these measurements can be made while you're drilling called Logging While Drilling (LWD).
Then you run a steel tube (called casing) into the hole and pump cement into the gap between the steel and the rock.
Then you may test the well- you blow a number of small holes in the steel tube to open up the well to the rock, and let the oil flow out, measuring the pressure all the time. Then you stop the flow and measure the rate the pressure builds up. From this you can determine other properties of the resevoir: permeability, the reseroir shape (occasionally) and so on.
That's exploration....
