Analytification of a variety I am working on understanding the article ANALYTIFICATION IS THE LIMIT OF ALL TROPICALIZATIONS by Sam Payne.
If $X$ is a closed subvariety of some torus $T\cong (K^*)^n$ where $K$ is an algebraically closed field. What do we mean by the analytification of $X$, which is denoted $X^{an}$?
 A: In Sam Payne's paper, $K$ is not just a field - it is one that is complete with respect to a non-archimedean norm. Consequently, any algebraic variety $X$ over $K$ has an associated $K$-analytic space $X^{an}$, in the sense of Berkovich. This is the non-archimedean analogue of the analytification of complex varieties: e.g. if $X$ is a smooth variety over $\mathbb{C}$, then $X(\mathbb{C})$ carries the structure of a complex manifold.
In the case when $X$ is an affine $K$-scheme of finite-type, it is straightforward to define the analytification $X^{an}$ as a topological space. Indeed, if $X = \textrm{Spec} (A)$ for some $K$-algebra $A$, then the points of $X^{an}$ are the multiplicative seminorms on the coordinate ring $A$ which extend the norm on the base field $K$. 
For each $f \in A$, there are evaluation maps $X^{an} \to \mathbb{R}_{\geq 0}$, sending a seminorm $|\cdot |$ to $|f|$. The analytification $X^{an}$ is equipped with the coarsest topology which makes these evaluation maps continuous. And in fact, this space is not as horrible as you may think! For example, if the variety $X$ is connected/separated/proper, then $X^{an}$ is a path-connected/Hausdorff/compact space. 
For a general $K$-scheme $X$ of finite-type, one can take an open affine cover and analytify each piece of the cover, and glue them together as analytic spaces. There is also a structure sheaf of analytic functions on $X^{an}$, but it is difficult to concisely explain.
In the specific example of the $n$-torus $X = \mathbb{G}_m^n$ over $K$, the analytification is then the set of multiplicative seminorms on the coordinate ring $K[t_1,t_1^{-1},\ldots,t_n,t_n^{-1}]$, with this topology of pointwise convergence. 
If you are new to the theory of Berkovich spaces, Sam Payne has a very nice survey paper which I highly recommend. If you are very brave, you can also take a look at Berkovich's book, where he develops the general theory of Berkovich spaces (but be warned, it is very terse!).
