Natural & important probability measures on $\mathcal{C}[0,1]$, in particular the Wiener measure

Regarding the identification between processes and measures:

Suppose $\{X_t : 0 \le t \le 1\}$ is a continuous stochastic process on a probability space $(\Omega, \mathcal{F}, P)$. Then there is a natural map $X : \Omega \to C([0,1])$ defined by $X(\omega)(t) = X_t(\omega)$. I claim this map is measurable with respect to the Borel $\sigma$-algebra on $C([0,1])$. Let $B = B(x,r)$ be an open ball in $C([0,1])$. Then $X^{-1}(B)$ is the set of all $\omega$ such that $|X_t(\omega) - x(t)|<r$ for all $t \in [0,1]$. By continuity, it is sufficient to check this holds for all rational $t$, and so we can write $$X^{-1}(B) = \bigcap_{t \in [0,1] \cap \mathbb{Q}} X_t^{-1}((x(t)-r, x(t)+r))$$ which is measurable in $\Omega$. Since the open balls generate the Borel $\sigma$-algebra, $X$ is measurable. Now let $\mu = P \circ X^{-1}$ be the pushforward of $P$ onto $C([0,1])$. $\mu$ is called the law or distribution of the process $\{X_t\}$.

Conversely, if $\mu$ is a Borel probability measure on $C([0,1])$, then $(C([0,1]), \mathcal{B}, \mu)$ is a probability space. For $t \in [0,1]$, define $X_t : C([0,1]) \to \mathbb{R}$ by $X_t(x) = x(t)$. Each $X_t$ is measurable (indeed a continuous linear functional) so $\{X_t : 0 \le t \le 1\}$ is a stochastic process, which is clearly continuous in $t$. It is also easy to check that the law of this $\{X_t\}$ is in fact $\mu$, since the map $X$ constructed above is just the identity in this case.

There isn't a measure on $C([0,1])$ which is quite as natural as Lebesgue measure on $\{0,1\}^\mathbb{N}$. In particular, there is no translation-invariant Borel probability measure (nor even one which is $\sigma$-finite). However, Wiener measure is a pretty important example, since it is "universal" in many ways. For instance, Donsker's invariance principle says it arises as the scaling limit of measures corresponding to random walks (with, say, piecewise linear interpolation). Karatzas and Shreve, Brownian Motion and Stochastic Calculus has the full statement (note that the Wikipedia page on Donsker's theorem describes a different result).

Wiener measure is an example of a Gaussian measure, of which there are many. I'll put in a plug for these lecture notes of mine where there is some more information on Gaussian measures.

Whenever $K(s,t)$ is a positive-definite function (for example, the Fourier transform of a positive measure, by Bochner's theorem), you can define an associated Gaussian process on $\mathcal{C}[0,1]$ with covariance function $K(s,t)$.

While the Wiener process is probably the best-known Gaussian process (to mathematicians, at least), there are many other examples. GPs are used quite extensively in machine learning.