"Transfinite" is a crappy word. It sounds cool, but it means just the same thing as "infinite" (namely "not finite"). Cantor devised it because he wanted a way to say "infinite" without offending certain theological sensibilities.
"In his 1883 paper entitled Grundlagen einer allgemeinen Mannigfaltigkeitslehre (“Foundations of a General Theory of Manifolds”) he introduced a distinction between two infinities, the transfinite and the absolute:
"Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
"The absolute infinite Ω, also introduced by Cantor, can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite."