How to name the edges?

begin{tikzpicture}
    
    node[shape=circle,draw=black] (0) at (0,0) {$0$};  
    node[shape=circle,draw=black] (4) at (6,0)  {$1$};   
    node[shape=circle,draw=black] (1) at  (3,-1) {$2$};  
    node[shape=circle,draw=black] (2) at  (3,2) {$3$};  
    
    draw (0) -- (1) node[pos=.5, below, sloped, rotate=0] {e1}; 
    draw (0) -- (2);  
    draw (0) -- (4);
    draw (0) -- (4); 
    draw (4) -- (2);  
    draw (2) -- (1);
    
    
end{tikzpicture}

It is not clear which names has particular path, but it is simple to correct/insert names. By using quotes library you can ad them as follows:

documentclass[border=3mm]{standalone}
usepackage{tikz}
usetikzlibrary{quotes}

begin{document}
    begin{tikzpicture}[
C/.style = {circle, draw}, %   C: circle
every edge quotes/.style = {auto, font=footnotesize, inner sep=2pt, pos=0.4}
                        ]
    begin{scope}[nodes={C}]
node (0)           {$0$};
node (4) at (6,0)  {$1$};
node (1) at (3,-1) {$2$};
node (2) at (3,2)  {$3$};
    end{scope}
%
path   (0) edge ["$e_3$"]  (2)
        (0) edge ["$e_1$"]  (4)
        (0) edge ["$e_2$"]  (1)
        (2) edge ["$e_4$"]  (1)
        (4) edge (2);
    end{tikzpicture}
end{document}

Addendum: This simple diagram can be easy drawn by employing tikz-cd package:

documentclass[border=3mm]{standalone}
usepackage{tikz-cd}
usepackage[low-sup]{subdepth}

begin{document}
    begin{tikzcd}[sep=huge,
                   cells={nodes={circle, draw, semithick, minimum size=1.1em, 
                          inner sep=2pt, outer sep=0pt}},
                   arrows = dash,
                   every label/.append style = {font=footnotesize, inner sep=1pt, pos=0.35}             
                   ]
    &   3 ar[dr,"e_{31}"]
          ar[dd,"e_{32}"]  &   
0 ar[ur,"e_3"]
  ar[rr,"e_1"]
  ar[dr,"e_2"]
    &                       & 1 
    &   2                   &
    end{tikzcd}