Here's a concrete example of an atomless measure.  Let $f \in L^1$ be an integrable function with total mass 1 (i.e. $\int_0^1 f = 1$).  Define $$\mathbb P(A) = \int_A f(x) ~dx$$ for any Borel set $A$.  It is a nice exercise to show that $\mathbb P$ is an atomless measure.  

Note:  $f$ is called the Radon-Nikodym derivative of $\mathbb P$ with respect to Lebesgue measure, and often written $f = \tfrac{d\mathbb P}{dx}$.  If a random variable $X$ has distribution $\mathbb P$, then $f$ is called its density function.