My understanding is that it just means fishing a dry line for steelhead and salmon. In the old days the silk line had to be greased to float. The method was an across presentation that had lots of mends to keep the big salmon flies skatingjust in or under the surface film. Others correct me if I'm wrong.
I think Randy is pretty much right. Some people think greased lining just means fishing a floating line, but it is more complicated than that from what I understand. I've never seen a demonstration of a greased line presentation, although I asked a local guide once and he sort of shrugged his shoulders. I think it is a bit of a dying art. There is a chapter on greased lining in Deke Meyer's Advance Flyfishing for Steelhead book, and there is also a book (probably out of print) of Bill McMillan articles, called Dry Flyfishing for Steelhead or something like that, that has a bunch of stuff on greased lining. As Randy says, it involves a cross stream cast and lots of mending. The fly is not dead drifted; you want the line, leader and fly extended and use mending to keep the fly slow and with as much of a cross stream presentation as possible. This is what I've read anyway - I can't claim I know how to do it in practice. Would love to learn though if I could find someone who actually knows how to do it.
As I understand it, in the modern era there is one definition of the word and it refers to fishing a dry line for steelhead and atlantic salmon. This is a simplified from the original definition from the days of silk lines, which refered to greasing the line and leader to present a wet fly just under the surface. The series of "low water" flies in salmon and steelhead patterns was designed to fill the need for a high floating wet fly.
The flies are presented in a standard 'steelhead swing' for summer fishing. But the followers of greased line fishing are in the minority, with the vast majority just swinging a smaller steelhead fly on a floating line in the summer.
For more info, try Trey Combs "Steelhead Fly Fishing", Jock Scott "Greased Line Fishing for Salmon and Steelhead" and various roderick Haig-Brown titles including Fisherman's Spring and the Western Angler.