Jack-up Drilling Rig
This type of rig is basically a hull, usually triangular, supported by 3 legs with a drilling package on top.

The legs pass through the hull of the rig and are 'jacked' up and down using very large motors.

When the rig is on location the bottom of the legs are touching the seabed. When the rig needs to be moved, the legs are jacked up.

Initially, this just causes the hull to move down the legs towards the water. When the hull reaches the water however it floats and the motors then start to raise the legs off the seabed, very slowly and up through the hull and into the air.

Tugs are then attached to the rig and it is towed to it's new location. When on it's new location, the legs are jacked down towards the seabed again.

Once they touch the seabed and settle the jacking action then starts to raise the hull out of the water and up the legs. Once a safe distance has been achieved between the base of the hull and the water, around 60-70' the rig is stabilised by taking on tons of water into 'preload' tanks and the rig is allowed to settle for a day to maximise stability. Drilling operations may then resume.

An oil company, ExxonMobil, Shell, BP etc would hire a drilling rig like this to drill exploratory wells in relatively shallow water to find oil or gas before constructing a production platform on the site after confirming the existence of commercially viable resources.