Rayon is the original artificial fiber, first developed by Georges Audemars in 1855. There are many different types of rayon, but all of them, but in layman's terms, all of them use some sort of chemical solvent to dissolve some source of cellulose (cotton lint, wood pulp, etc.) into a mass of goo, which is then extruded into fibers and spun into yarns. It's not like polyester, which often has difficulty breathing and can melt at high heat. Since it's cellulose, it behaves in many ways like cotton (another fiber that's largely cellulose) -- it can breathe, it can absorb water well -- but since it's an engineered fiber, it can often perform better than cotton in many ways (depending on what it was engineered to do). Today, there are many different kinds of rayons, each engineered for different uses. Lenzing Fibers, an Austrian company, is one of the leading producers of rayons, and their two flagship products are Tencel and Modal. Tencel is Lenzing's trade name for lyocell, a rayon made from wood pulp. Modal is made from the pulp of beech trees. It's extremely soft, and it's more hydroscopic than cotton (which means that it absorbs water more readily than does cotton). More than that, it also takes dyes readily and is very colorfast, meaning that bright reds like the one in the Zimmerli undershirt pictured above don't fade with washing, as they would if the fabric were made from cotton. Modal is widely used as a blending fiber with cotton, and it's also a standalone. I have seen sheets made both from 100% Modal and cotton/Modal blends, and everybody and his uncle (including Target and the Gap) is selling Modal or Modal-blend underwear and sleepwear.
A step up from regular old Modal is MicroModal, also produced by Lenzing. As far as I can tell, the primary difference between Modal and MicroModal is the fineness of the fibers produced, with the fibers in MicroModal obviously being finer. As far as I can tell, MicroModal is only available in knit fabrics (I have seen some woven Modals), and those knits are overwhelmingly used for underwear, including Zimmerli's Pureness line (which blends MicroModal with a small amount of Lycra). It's very expensive, but it's also incredibly soft. And it retains that softness through many, many washes. Good stuff, and I wish that someone would make a woven fabric from MicroModal -- I think it would make a great shirt.