As a ring, a standard affinoid algebra (or Tate algebra) $T_{n,K}$ is the subring of the ring of [[formal power series in $K[ [x_1, \ldots, x_n] ]$ consisting of all strictly converging series $c= \sum_I c_I x^I$, that is such that $|c_I|\to 0$ as $I\to \infty$.

There is a Gauss norm? on such series $\|\sum_I c_I x^I \| = max\{|c_I|\}_I$. This is indeed a norm making $T_{n,K}$ into a Banach $K$-algebra of countable type.

An affinoid algebra is any Banach algebra which can be represented in a form (Tate algebra)/(closed ideal).