What is a white noise process? What is the best way of defining white noise process so it is intuitive and easy to understand?
 A: A white noise process is one with a mean zero and no correlation between its values at different times. See the 'white random process' section of Wikipedia's article on white noise. 
A: A white noise process is a random process of random variables that are uncorrelated, have mean zero, and a finite variance. 
Formally, $X(t)$ is a white noise process if $$E(X(t)) = 0, E(X(t)^2) = S^2\text{, and } E(X(t)X(h)) = 0 \text{ for } t\neq h\text{.}$$
A slightly stronger condition is that they are independent from one another; this is an "independent white noise process."
A: I myself usually think of white noise as an iid sequence with zero mean. At different times values of the process are then independent of each other, which is much stronger requirement than correlation zero. What is the best with this definition that it works in any context. 
Side note. I only explained my intuition, the correct definition of white noise is given by @onestop. The definition I gave is mathematically defined as white noise in strict sense.
