Offshore is a term used within the oil and gas and supporting offshore industry to refer to work locations that are at sea beyond the sight of land.
In these industries, locations for installations, fields, and facilities are separated into three categories:
Onshore is anything on land (including beaches and islands). Inshore refers to locations at sea, but limited to areas within sight of land.
So offshore is everything else -- at sea, out of sight of land.
When we talk of offshore facilities and offshore workers, we are referring to platforms and oil fields located at sea, out of sight of land, and the people who work at these locations.
The Offshore Industry
The offshore industry usually describes the support industry for the oil and gas companies that operate offshore. This includes the companies that conduct periodic inspections of the facilities, the surveyors that ensure everything is placed in the right locations, and the specialists who build and maintain the equipment used there.
It is usually referred to separate from the oil and gas industry, which refers to the companies (and their direct employees) who deal with the drilling, production, storage, and offtake of the petroleum products.
Interestingly, in this industry mariners and sailors do not generally fit into the category of 'offshore worker'. This term is reserved for the workers who are offshore specifically to work with the oil and/or gas deposits. Seamen (referred to as marine crew), must meet the qualifications associated with the maritime industry, they are generally employed through the vessel on which they are assigned, rather than through the oil and gas companies conducting the work.
Their working conditions are also different -- most notably, offshore workers generally work on a rotation basis (eg. four weeks on (hire), two weeks off (at home)), whereas marine crew endure the solitude prescribed by a life at sea.