Let's start by paraphrasing the term "irrational number": a number whose decimal expansion neither terminates nor repeats (after the decimal).
sqrt(2) is irrational: 1.41421356... on forever
Phi (the golden ratio) is irrational: 1.61803398...
Pi is irrational: 3.1415926535...
This I'm sure you already new but it helps to explain a transcendental number a bit better.
A transcendental number is a number that cannot be expressed with only a finite number of algebraic operations.
So sqrt(2) is irrational but not transcendental because obviously it can be expressed as "sqrt(2)", and so the same goes for the golden ratio which is expressed as (1 + sqrt(5))/2.
But pi cannot ever be expressed as say... (a + sqrt(b))/c where a, b, and c are integers.
This is equivilant to saying pi cannot be a solution to a polonomial of degree n with integer coefficients.
So goes for the number e (2.718281828) which can be expressed by a continuing series but not algebraicly.
lim n -> infinite 1 + 1/2! + 1/3! + 1/4! ... 1/n! = e
pi can also be expressed by many such continuing series, continued fractions, and so on.
A transcendental equation is one which cannot be pinned down to a finite number of terms or operations. Take e^x (e raised to the x'th power) = 1 + x + x^2/2! + x^3/3! + x^4/4! ... x^n/n! ("... x^n/n!" implies "on to infinite")
Other transcendental equations include sin(x) and the other trig functions, log(x) or ln(x), the Riemann zeta function (a very interesting and mysterious equation), etc.
Tetration is simply the next logical step in the progression of algebraic operations.
Multiplication = repeated addition
2x means x added to itself two times.
3x means x added to itself three times.
Nx means x added to itslef N times.
If N happens to equal x, we then express this as x^2
Exponentiation = repeated multiplication
x^2 means x multiplied by itself twice.
x^3 means x multiplied by itself three times.
x^n means x multiplied by itself n times.
If n happens to equal x (x^x), we then express this in the notation for tetration (difficult
to use in this forum). I guess x^^n would have to do...
Tetration = repeated exponentiation
Quintation = repeated tetration
Sextation = repeated quintation
and so on...