I am reading Ruzhansky's textbook on pseudodifferential operators and came across this passage: 
I have never seen a sentence before that says this sequence of functions converges pointwise, uniformly in $\epsilon$. In the context of differential and pseudo-differential operators, what does this usually mean?
For a given $\varepsilon$, $a_\varepsilon$ is a function which approximates $a$. Each of these are themselves functions of $(x,\xi)$.
That $a_\varepsilon$ converges pointwise to $a$ as $\varepsilon \to 0$ means that $a_\varepsilon(x,\xi) \to a(x,\xi)$ as $\varepsilon \to 0$ for each fixed $(x,\xi)$.
That $a_\varepsilon$ converges uniformly to $a$ in $0\leq \varepsilon < 1$ means that the error function $|a - a_\varepsilon|$ (itself also a function of $(x,\xi)$) can be bounded by a function $\delta(\varepsilon)$ (not depending on $(x,\xi)$) such that $\delta(\varepsilon) \to 0$ as $\varepsilon \to 0$. They also make the choice to restrict $\varepsilon$ (and thus the domain of $\delta(\varepsilon)$) to $[0,1)$, probably for convenience.