The concept of analytic spectrum is a realization of the concept of spectrum (geometry) in the context of non-archimedean analytic geometry. Given an affinoid algebra $A$ over a non-archimedean field, then the concept of spectrum of a commutative ring whose points are prime ideals/maximal ideals does not produce a sensible space that admits analytic geometry. Rather, instead of regarding points of the spectrum as ring homomorphisms to $A \to \mathbb{R}$, the analytic spectrum instead takes points to be multiplicative seminorms ${\vert -\vert} \colon A \to \mathbb{R}_{\geq 0}$ bounded by the norm on the given field.
If $Spec_{an}(A)$ is the set of all such multiplicative seminorms, then for $x \in Spec_{an}(A)$ a point one writes the corresponding seminorm as ${\vert - \vert}_x$ and thinks of it as being the norm on the function algebra on $Spec(A)$ which is given by “evaluating functions at $x$ and then applying the field norm to that”.
One turns this $Spec_{an}(A)$ into a topological space in the usual way by choosing the weakest topology such that under this assignment the original elements of $A$ become continuous function on $Spec_{an}(A)$.
Globalizing this analytic spectrum construction leads to the concept of Berkovich analytic space.
For $A$ a normed ring, its analytic spectrum or Berkovich spectrum $Spec_an A$ is the set of all non-zero multiplicative seminorms on $A$, regarded as a topological space when equipped with the weakest topology such that all functions
of the form
for $a \in A$ are continuous.
If $A$ is equipped with the structure of a Banach ring, one takes the bounded multiplicative seminorms.
So a point in the analytic spectrum of $A$ corresponds to a non-zero function
to the real numbers, such that for all $x, y \in A$
${\Vert x \Vert} \geq 0$;
${\Vert x y \Vert} = {\Vert x \Vert} {\Vert y \Vert}$;
${\Vert x + y \Vert} \leq {\Vert x \Vert} + {\Vert y \Vert}$
and boundedness means that there exists $C \gt 0$ such that for all $x \in A$
(e.g. Berkovich 09, def. 1.2.3)
For $k$ a field and $k[T]$ the polynomial ring over $k$ in one generator,
is the analytic affine line over $k$.
If $k = \mathbb{C}$, then $\mathbb{A}_k = \mathbb{C}$ is the ordinary complex plane.
The notion originates in
Introductory exposition of the Berkovich analytic spectrum includes
Sarah Brodsky, Non-archimedean geometry, brief lecture notes, 2012 (pdf)
Scott Carnahan, Berkovich spaces I (web)
Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)
Frédéric Paugam, section 2.1.4 of Global analytic geometry and the functional equation (2010) (pdf)
Vladimir Berkovich, section 1 of Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)