These are troublesome terms, used mainly to describe the cap surface of a mushroom. They all have something to do with the surface being broken up into many smaller areas, usually by cracks in its surface. (If you don't care about the background, and just want a quick definition (that's probably most non-professionals), just skip down to the pictures).
Some of the terms, such as areolate and tesselate, are named for the areas; others, such as rimose and rimulose, are named after the cracks. Some of these terms would seem to be redundant (especially the compound ones), since one certainly can't have a network of cracks without them marking out areas in between them, and likewise its difficult to have division into areas without something acting as a divider.
In particular, there has been a specious division in several works between the terms rimose and rimose-areolate, where rimose has been defined to mean "irregularly cracked, but with the cracks not crossing one another", with rimose-areolate reserved for when the cracks cross one another, creating our friends, the "areas". I don't think that this usage existed before Snell & Dick(1957) , and I attribute it to Snell sitting down and trying to imagine what these terms would mean if there was a real reason for them all to exist and they all made sense.
In fact, by this definition, I know of no mushroom that has a rimosecuticle as a diagnostic characteristic. Many mushrooms have a cuticle that starts cracking in the sun, or as the mushroom dries out. This cracking usually starts at the edge of the pileus, and progresses inward; it very quickly reaches the stage where the cracks cross one another and mark out areas. Thus, books that use Snell's definitions (and you can recognize them by this) end up calling everything rimose-areolate, because the only available use for " rimose" is the brief transitional zone in between the cracked and non-cracked areas.
I am fairly certain, however, that the basic terms ( rimose and areolate) were coined first, to designate the most common conditions, and that the terms derived from them came later. I have therefore used these terms in this website in the following sense:
A rimose surface is divided by cracks into discrete areas, as in Boletus fraternus. This is by far the most common condition, and if you are looking at any of the listed terms used outside of this website, odds are that this is what they really mean. Areolate, as far as I'm concerned, is a synonym.
Crustose will denote conditions as in Russula crustosa and Russula virescens, where the colored layer of the cuticle has been stretched and broken into pieces as the cap grows, but the cuticle as a whole is still intact. The "areas" may be thicker than the rest of the cap, but the height of a cell or two, but there are no real "cracks" between them. Another way of thinking of this is that there's a very thin "crust" layer on the cap; as the cap expands, the crust gets broken up into smaller pieces, but the underlying tissue remains whole.