TENCEL is made of wood pulp from sustainably certified beech and eucalyptus trees, which is made into fibers, that can be spun. The composition of natural fibres from the wood, makes Tencel utterly comfortable to wear. It is soft and breathable, and it has a really good ability to absorb moisture and let it go again. The fibres are naturally anti-bacterial, which means less need for washing. The natural fibres are anti-static and doesn’t cling to the body.
MODAL shares a lot of similarities with TENCEL when it comes to softness, comfort, breath-ability and moisture absorption. The Modal fibres are also derived from sustainable wood sources (mostly beech wood) that are harvested from certified and controlled sources in natural forests and sustainably managed plantations.
You can call MODAL the 2nd generation of the regenerated fibres (viscose/rayon is 1st and TENCEL is 3rd), and the story of an environmentally responsible production process, where wood pulp is turned into cellulose fibers with high resource efficiency and low ecological impact is pretty much the same for Modal and TENCEL fibres. The main difference for you is that Modal has a slightly more delicate touch and feel. It feels softer and more 'milky', and it is often made into lighter and thinner fabrics compared to TENCEL.
CUPRO is a regenerated fiber made from cotton linter (the ultrafine, silky fibers that stick to the seeds of the cotton plant after it’s been ginned). Cotton linter is normally a waste product from the cotton production because it can't be used in the fiber production. Although Cupro is derived from cotton, it is very fine, and does not have the non-uniformity of thickness found with natural fibers. Cupro is anti-static, breathable, has great color retention and it's gentle on skin. It has a very smooth surface and a low coefficient of friction, making it slide smoothly for a light, comfortable fit. Cupro has a unique 'peach-skin' finish, and it often has the look and feel of a silk.
For inspiration for what you can sew with Tencel, have at look at this blog post.