Labeling an edge in TikZ

One way is to use the quotes library. (If you use the babel package, you also need to load the babel library.)

documentclass[tikz,border=3mm]{standalone}
usetikzlibrary{quotes}
begin{document}
begin{tikzpicture}[node/.style = {circle,draw}]
  node[node] (B) {B};
  node[node] (A) [below of=B, xshift = -20mm] {A}
    edge [-,"1"] (B); % would like to label this edge "1"
  node[node] (C) [below of=B] {C}
    edge [-,"1"] (A); % would like to label this edge "1"
  node[node] (D) [below of=C] {D}
    edge [-,"1"] (A); % would like to label this edge "1"
  node[node] (E) [below of=B, xshift =  20mm] {E}
    edge [-,"1"] (B) % would like to label this edge "1"
    edge [-,"1"] (C) % would like to label this edge "1"
    edge [-,"1"] (D); % would like to label this edge "1"
end{tikzpicture}
end{document}

With use of the TikZ libraries automata, positioning and quotes, nodes and paths are drawn separate, in positioning of nodes is used syntax determined in positioning library:

documentclass[tikz,border=3mm]{standalone}
usetikzlibrary{automata,
                positioning,
                quotes}
begin{document}
    begin{tikzpicture}[
node distance = 15mm, on grid,
every state/.style={semithick, minimum size=1em, inner sep=2pt},
every edge/.append style = {-},
every edge quotes/.append style = {auto, font=footnotesize, inner sep=2pt}
                        ]
    begin{scope}[nodes={state}]
node (C)       {C};
node (A) [ left=of C] {A};
node (B) [above=of C] {B};
node (E) [right=of C] {E};
node (D) [below=of C] {D};
    end{scope}
%
path (A)   edge ["1"] (B)
      (B)   edge ["1"] (E)
      (E)   edge ["1"] (D) 
      (D)   edge ["1"] (A)
% 
      (A)   edge ["1"] (C) 
      (C)   edge ["1"] (E); 
    end{tikzpicture}
end{document}