TacBeezy

So land and sea operation start a little bit differently but in the end, they are the same process. To start, you need to spud the well. This is the start of the well and on land often consists of driving a large diameter (36") conductor casing (metal pipe) into the ground. This basically starts the well and will lend the well system structural stability later on in the process. After the conductor casing is in, drilling as you would think of it begins. Most land rigs in the US now use top drives which are huge electric motors that actually rotate the entire drill string much like a giant handheld electric drill. The drill string is made up of a bit on the bottom and followed by pipe on the top. As the drill string is rotated, they use the weight of the drill pipe above to apply pressure to the bottom of the hole. As pipe is pushed down into the hole, the rig crew attaches additional pipe on the rig floor (the part you see above ground with the high tower, or derrick, above it). This is how they drill into the ground.

Now that we have the basic drilling process down, we can worry about some other stuff like friction, what to do with the cut rock, and actually finding hydrocarbon reserves. Friction down hole is immense and thus creates a TON of heat. Because of this, and the need to remove rock cuttings from the bottom of the hole, we use drilling mud. This can range anywhere from air to water to synthetic oil based muds with very specific rheological properties. This mud is contained in tanks or in pits on the ground near the rig. It is then pumped through the drill pipe and out of nozzles in the bit. This high pressure spraying will clean the chunks of rock called cuttings away from the bit and out of the well. The cuttings are separated from the mud on surface and the mud is recycled.

Now we have drilled effectively and gotten all of our cuttings out from the annulus around the drill string. Once we hit hydrocarbon, we run into the issue of poor pressure. Hydrocarbons are contained in pores within certain porous formations. Due to the pressure of the overlying rock, these fluids have a tendency to flow out of the formation. If this flow is uncontrolled, it can lead to a blowout like OP linked a video to. We use the drilling mud to control this migration of fluid. Mud engineers make the mud at a certain density such that the hydrostatic pressure of the mud is enough to prevent the fluid in the pores from coming out.

That is the basic drilling process. Now, depending on how deep you need to drill, how deviated the wellbore is, and the geological features around the well; some additional structure may be needed. Wells are often drilled in sections of decreasing diameter until the target depth is reached. Upon the completion of each section, straight metal casing is run to line the hole. Then, for additional stability, cement is pumped in to fill the gap between the casing and the borehole wall. This leaves us with a secure, stable hole drilled down to the area of the hydrocarbon formation. To get the hydrocarbon out, perforating guns can be run down and blown to produce holes in the casing and cement.

This is a very high level overview and I have glossed over a lot. Comment if you have any more questions or want more specifics on any section.

I work as a drilling and measurements engineer on deepwater rigs in the Gulf of Mexico... If you have questions about drilling specifics, I can almost definitely answer them.