Offshore rigs could be classed into two types - bottom supported, and floating rigs (I made the first category up, maybe there's an official name for it but I don't know it and 'bottom supported' will do).
There are offshore rigs available to drill over any depth of water, from shallow lagoons you could swim to the bottom of to harsh icy seas which are 10,000ft deep.
Jack-ups and platforms can be classed as bottom supported rigs - they both have legs which attach the rig to the sea (or lake) floor, thus stabilising them and allowing them to be higher above any potentially damaging waves.
Semi-submersibles and drill ships are the most common type of floating rigs.
Offshore rigs (also known as modular offshore drilling units, or MODUs), whether jack-ups, semi-submersibles, or drill ships, usually have a number of common features which aren't found on land rigs, such as:
- Accommodation - dining facilities, cabins, gyms, medical rooms, etc. will usually be integral to the offshore unit - it needs to be self sufficient.
- Communications - satellite link, radar, sonar, radio.
- ROV (remotely operated vehicle) - An unmanned submarine deployed from the rig, used to monitor and perform subsea operations (controlled by the ROV team on the rig, they have sensitive control manipulators and can be used for lots of jobs - dredging, performing BOP and wellhead operations, surveying the seabed).
- Bulk loading system - Water, fuel, and drilling additives (like bulk (dry powder) cement, or mud and mud chemicals) need to be loaded from supply vessels which come alongside, and the bulk loading system comprises hoses which take onboard these supplies and direct them to the correct tanks and silos.
- Deck cranes - offshore rigs usually have several permanent cranes to handle cargo from supply vessels, and to lift items to and from the drill floor.
- Sea chest - a pump and series of pipes which allows seawater to be brought up to the rig to be used to make e.g. cement slurries, and to drill with if required.
- Heli deck - often the quickest, easiest and safest way to get to a rig is via helicopter. The heli deck needs to be certified that it's capable of having a helicopter land on it. Boat transfers, when personnel are mobilised to and from the rig in specially made contraptions suspended from a crane, are also available and in common use should helicopters not be available.
See more about the types of offshore rigs commonly used: