As with lots of business speak, the term has been diluted through over use and misuse. But basically, a line manager sits in the "line" of command, i.e. it is someone with operational responsibility over a part of the business. Line managers therefore appear in all bands, from junior managers to the most senior.

Line management comes with responsibility over staff executing the operations and in doing so managing other people's work. Your line manager is therefore your direct boss as opposed to others, including more senior ones, in the "line" of command.

For example, when I was a director, I would "line manage" the departmental heads. They in turn would "line manage" their staff, which include other managers who again in turn oversaw more junior people etc. I would be the senior manager for the whole department, but only the line manager for the departmental heads at the level immediately below me. My line manager was a member of the board. The chief exec's line manager was the chairman.