I'm trying to describe to my students what level designers do. The first thing to say, of course, is "it depends"--depends on what the company expects the level designer to do, and what is "farmed out" to someone else.
Nonetheless, the analogy to a Dungeons and Dragons or other paper RPG referee is good, though a great many younger people don't seem to be familiar with (non-video) D&D these days.
Fundamentally, level design is a limited, concentrated form of game design. The core mechanics of the game are already determined. The level designer is using them to create an episode that will have good gameplay, that will entertain in various ways. Gameplay always involves challenges and actions to meet those challenges, of course. This also involves goals, ways to achieve the goals, paths (such as corridors and rooms), appearances, and behavior of NPC's and opposition (scripts to do better than the game AI can do on its own).
In D&D the "DM" or DungeonMaster starts with the "core mechanics" of D&D and fleshes out adventures. The adventure, analagous to a video game level, usually involves a goal of some kind, if only to "wipe out the badguys". The DM may have particular methods in mind whereby the players can achieve the goal, or he may simply set up a situation and trust the players to creatively find ways to achieve the goal. In level design, playtesting will show whether creativity can prevail; in home-made D&D adventures there is no playtesting, so the DM must be more careful. But D&D adventures that are published are certainly playtested.
A published D&D adventure includes all--well, most--of the information a referee needs to run the adventure. The video game level includes everything needed for the player(s) to play the adventure--er, level.
So the level designer must specify and perhaps place (though probably not make) the graphics, map out all the paths and alternatives the player(s) can pursue, place the opposition (monsters or otherwise), script the conversations, specify the goal and how player(s) find out what that goal is, specify exceptions to the normal core mechanics, and all the other things that are required for the "adventure" episode.
"The worst form of inequality is to try to make unequal things equal." -- Aristotle
"Always do right--this will gratify some and astonish the rest."Mark Twain
"A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away." Antoine de Saint-Exup'ery
"Not everything that can be counted counts, and not everything that counts can be counted." Albert Einstein
"Make everything as simple as possible, but not simpler." Albert Einstein