What should be understood by "refrigerate until firm, 2 hours or overnight"?

When I've seen that step in cookie recipes, generally there's no way you can over-firm in the fridge; if you're doing all the steps on the same day, it'll take two hours to set, but if you're trying to prep the night before, you can break for the night at this step.

Generally you'll have a wet, sticky, soft dough you can't cut into shapes, and after a trip to the fridge, it'll be firm enough to hold a shape when you go at it with a cookie-cutter (or roll it into a pinwheel and slice, or whatever shaping is happening). It's basically letting the butter go back from room-temperature-butter consistency to fridge-butter consistency.

When cookie recipes ask to be refrigerated for a period of time or overnight, it is about more than simply the final temperature. You cannot, for example, chill the dough in an ice water bath (sealed in plastic, for example, to keep it from getting waterlogged) to get it to a lower temperature faster. The outcome is about time as well as about temperature. In fact, in some ways time is more important, and the refrigeration is to keep it safe (since cookie dough usually has raw eggs).

While the final temperature does effect the firmness of any fats or shortenings in the cookie directly, and that effects how easy they are to roll out, another major issue is hydrating the flour.

The longer the rest, the more time there is for the starch granules in the flour to absorb water, which will make your dough more cohesive (stays together well) and workable. This is especially important for rolled out or sliced cookies, which I suspect your particular recipe is.

The texture of rested dough will be putty-like and pliant, It will not crack easily when pressed, rolled, or pinched.

The majority of this effect takes about four hours—I am surprised your recipe specifies only two hours. Overnight is even better, although not usually necessary.

Note: Most refrigerators would be approximately: 40°F / 4.5°C, since you asked what temperature is expected.

See also: What does an overnight chill do to cookie dough, that a 4 hour chill doesn't?

• Firm here means that it should not be supple, nor too hard to be easy to cut. It will be sustaining shape. If you have a round cylinder of cookie dough, cutting it should keep the cylinder in the same shape rather than flattening the sides where you are holding the dough and flattening the dough around the cut (dough too soft).

• 2 hours or overnight means at least two hours. 2 hours to get some of the benefits of firming the dough; more could potentially give you a better result; the dough is firm all the same after the night.

• as mentioned by @SAJ14SAJ, resting the dough may give you other properties such as different elasticity and cohesion. I don't know what the author of your recipe had in mind; he may or may not look for those properties to change.