I recall that a seminorm is basically a norm that is not necessarily positive definite.
Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with this distance :
$$d(x,y):=\sum_{n\geq0}\frac{1}{2^n}\frac{p_n(x-y)}{1+p_n(x-y)}$$
Then $(E,d)$ is complete. Let $(E',(p'_n)_{n\in\mathbb{N}})$ be another Fréchet space. We all know the definition of a uniformly continuous function between two metric spaces. But what I want to know is a way to characterize uniform continuity of a function $f:(E,(p_n)_{n\in\mathbb{N}})\rightarrow(E',(p'_n)_{n\in\mathbb{N}})$ in terms of the seminorms $(p_n)_{n\in\mathbb{N}}$ and $(p'_n)_{n\in\mathbb{N}}$ instead of the associated distances $d$ and $d'$. I can't find an answer anywhere in any book that I've read that discusses Fréchet spaces.
P.S: Maybe completeness is an unnecessary condition. I added it just in case we need it.
You should look into uniform spaces. These are generalizations of metric spaces, and Fréchet spaces are examples of uniform spaces, so are any topological vector space. Some possible references for uniform spaces is Bourbaki's General Topology or Kelley's General Topology. However, since you are probably more interested in the uniform structure of topological vector spaces these references are maybe to general. In that case I suggest Schaeffer and Wolff's Topological Vector Spaces or Narici and Beckenstein's Topological Vector Spaces, both of which cover the uniform properties of topological vector spaces.