Stabiliser is the backing that takes the load of the stitching so the fabric does not have to. Getting it wrong shows up as puckering, wavy lettering, or a design that looked fine in the hoop and went crooked after the first wash.
Stays on the garment for life; you trim the excess and the rest remains behind the stitching. Use it on anything that stretches: t-shirts, polos, jersey, sweatshirts, performance wear. This is the single most common fix for lettering that goes wavy after a wash, because a tear-away lets the knit move again as soon as it is removed.
Torn off after sewing. Fine on stable woven fabric where the fabric itself holds the design: caps, denim, canvas, tote bags, twill. Cheaper, cleaner on the back, and completely wrong under a t-shirt.
Dissolves in water. Two uses: as a backing for lace and free-standing pieces where no fabric remains at all, and as a topping on the front of pile fabric so stitches sit on top of the loops instead of sinking between them. A topping on a towel is often what makes lettering readable.
Fabric and stabiliser go in the hoop as one. Hooping only the stabiliser and floating the fabric on top is a common shortcut and a common cause of gaps between an outline and its fill, because the fabric can creep under the needle even though the backing cannot.
Hoop firmly, not tightly. Fabric stretched drum-tight in the hoop relaxes when it comes out, and the design shrinks with it.
A cut-away. Jersey stretches in every direction and a tear-away lets it move once the hoop comes off, which is what makes lettering on a t-shirt look wavy after the first wash.
Both together. Hooping only the stabiliser and floating the fabric on top lets the fabric shift under the needle, which is a common cause of gaps between an outline and its fill.