None of our references explain concisely what Soar does, and some (e.g. Winston) only explain a tiny part of it. So I've put together the following sections from a number of different sources. This was not, incidentally, particularly easy.
In the problem-space paradigm, our problem consists of a number of states, and we have various operators for converting one state into another.
Example: the eight-puzzle. This is a 3 by 3 frame which can hold eight tiles, numbered one through eight. Instead of a ninth tile, there's a gap. The problem is to move the tiles so that they end up in a specified configuration, e.g. all the numbers in order. The states are the various different configurations of the puzzle. The operators are actions like ``move the gap right'', ``move the gap up'', etc.
Example: a robot in a room with a box, e.g. the monkey and banana puzzle. The states are the relative positions of robot, box, and other objects. The operators are actions like ``robot, move to box'', ``robot, push box'', etc.
Note. Some problems can have many different ways of describing operations. E.g. in the eight-puzzle, the operations could be more specific - ``move tile 9 left'', etc.