A drilling rig was a mechanical device that was used to drill deep holes into the ground.
The Enterprise NX-01 was equipped with a portable drilling rig. In 2151, Malcolm Reed and Travis Mayweather used the device to take a core sample from Archer's Comet which was rich in eisillium. The drill hit a layer of magnesite which broke the drill bit, but the away team was able to replace the bit and successfully take a core sample. (ENT: "Breaking the Ice")
In the alternate reality, a drilling rig was the primary weapon of Nero's ship, the Narada, which was actually a mining vessel. Nero used the drilling rig to drill down to the core of Vulcan, then dropped a "red matter" bomb down the hole. The red matter created a singularity at the center of the planet which would consume it. He tried the same thing against Earth, but the drilling rig was destroyed by Spock after he commandeered the Jellyfish, before it reached Earth's core. (Star Trek)