Conflicts between Circuitikz and Babel

The usual problem with babel shorthands. You cure it with \usetikzlibrary{babel}.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[spanish]{babel}

\usepackage{circuitikz}
\usetikzlibrary{babel}

\ctikzset{v/.append style={/tikz/american voltages}}

\begin{document}

\begin{circuitikz}[american voltages]
  \draw (0,0) to [sV]
        (0,2) to [R, v^>=$v_1$]  %%This is the problem.
        (3,2) to [C]
        (3,0) -- (0,0)
  ;
\end{circuitikz}

\end{document}

As you note in the question, the problem has nothing to do with biblatex but with the spanish option of babel. This is due to the > shorthand defined in the Spanish babel. If you don't need or use the shorthands at all then you can turn them off globally with the package option es-noshorthands.

\usepackage[spanish,es-noshorthands]{babel}

Alternatively if you need the shorthands, you can disable the offending one (>) within the circuitikz environment:

\usepackage{etoolbox}
\AtBeginEnvironment{circuitikz}{\spanishdeactivate{>}}

Here's a complete document:

\documentclass[10pt,twocolumn]{article}    

\setlength{\columnsep}{20.0pt}
\usepackage[spanish]{babel} 
% or if you don't need shorthands at all:
%\usepackage[spanish,es-noshorthands]{babel}
\usepackage{circuitikz}
    \ctikzset{v/.append style={/tikz/american voltages}}   
\usepackage{etoolbox} 
\AtBeginEnvironment{circuitikz}{\spanishdeactivate{>}}
\begin{document}

\begin{circuitikz}[american voltages]
      \draw
      (0,0) to[sV]
      (0,2) to[R, v^>=$v_1$]  %%This is the problem. 
      (3,2) to[C]
      (3,0) -- (0,0)
      ;
    \end{circuitikz}



\end{document}