In (one formulation of) the traveling salesman problem, you're trying to find a path that's as short as possible while still hitting all nodes.
An approximation algorithm will give you a path that hits all the nodes but is not necessarily the shortest. You know, however, that any optimal solution can't be longer than the approximation - the optimal solution has to be at least as short as every solution, including the approximation.
Since the optimal solution can't be longer than the approximation, the approximation is an upper bound: "I've found a path of length 5. That may not be the shortest path, but I know that a shortest path can't be longer than 5."