the short way of putting it I guess is that algebraic numbers are the solutions to some polynomial equation with integer coefficients. Rational roots of rational numbers fall into this category as well as rationals, integers, etc.
A transcendental real number is a real number that isn't algebraic.
A transcendental function is one cannot be expressed as a polynomial with algebraic coefficients.
I updated the definition of these. The one I posted before wasn't really correct. I bet one of the real mathematicians on here can give you a better definition yet.
The hyperbolic trig functions? Yes, these are transcendental. They are made up of exponential functions which are transcendental.