Brownian motion is gaussian process, but also integral of gaussian process?

Question

My course in stochastics has just defined a Brownian motion (Weiner process) $B: \Omega \to C[0,1]$ as a distribution over $C[0,1]$ functions. Equivalently, we can interpret it as a family of random variables indexed by $t \in [0,1]$ (that satisfy certain conditions, such as independence of increments). I see on Wikipedia that there are a number of equivalent definitions; one is defining a Brownian motion to be a Gaussian Process with covariance funciton $K(B_s, B_t) = \min(s, t)$. However, I see also on Wikipedia that it can be characterized as the definite integral of a white Gaussian process. My questions are 1) could someone write the latter integral formally, defining each symbol carefully, and 2) how can this stochastic process both be a GP itself and a definite integral of a GP?