There are infinitely many roots of $\zeta(s)-a=0$ for every complex number $a$. When $a\ne 0$, these are called "$a$-values" and there is a whole chapter discussing their distribution in Titchmarsh's book on the zeta-function. Selberg also discusses $a$-values in his (now famous) paper "Old and new conjectures and results about a class of Dirichlet series" where he defines the Selberg class.
Here is an overview of some of the important results:
1) There are $\frac{T}{2\pi}\log T + O(T)$ $a$-values of $\zeta(s)$ in the strip $0<\Im s\leq T$.
2) Like the zeros of $\zeta(s)$, Levinson proved that $a$-values cluster near the half-line. That is to say, almost all $a$-values are arbitrarily close to the half-line.
3) Unlike the zeros of $\zeta(s)$, there are provably a lot of $a$-values away from the half-line (though not a positive proportion). Namely, there are $\gg T$ roots of $\zeta(s)=a$ for $a\neq 0$ in any strip region $A\leq \Re s \leq B$ and $0<\Im s\leq T$ where $A\in (1/2,1)$ and $A$ strictly less than $B$. This is proved in Titchmarsh's book. On the other hand, standard zero-density estimates for the zeta-function tell us that there are $o(T)$ zeros in such a region. Some have suggested that this is evidence for the Riemann Hypothesis.
