Placing node right after end of path in Tikz

You can put the node at pos=1 and then left (or right, depending on where) align it:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[every node/.style={draw=red},
    ]
         \draw (0,0) -- (1,0) node[right]{a} ;
         \draw (1,0.5) --node[pos=1, left]{b}  (0,0.5) ;
         \draw (0,-1) -- node[pos=1,  right]{c} (1, -2)  ;
         \draw (0,-1.3) -- node[pos=1,  sloped, right]{d} (1, -2.3)  ;
         \draw (0,-2) .. controls (0,-1) and (0.5,-1.5) .. node[pos=1,  right]{e} (1, -1)  ;
\end{tikzpicture}
\end{document}

One can get somewhat close using the decorations.markings library. The code below makes use of the fact that a markings decoration has an internal coordinate system where the x-axis is parallel to the path, so I made a short path along the y-axis, and placed an auto node along it.

It's not perfect, and I can't guarantee that it will always work. For the bent to paths in the example below, I had to rotate the nodes by -90 degrees, and I'm not exactly sure why. You can also see that the position of the nodes is perhaps not perfect.

\documentclass[border = 2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[
  endnode style/.style={},
  endnodestyle/.style={endnode style/.append style={#1}},
  endnode/.style={
    decoration={
     markings,
     mark=at position 1 with {
       \path (0,-1pt) -- node[auto,swap,endnode style] {#1} (0,1pt);
     }
    },
    postaction={decorate}
  }
]
\foreach \Ang in {0,45,...,350}
{
  % works fine for straight lines
  \draw [endnode=a] (0,0) -- (\Ang:2cm);
  % needs a rotation for these curved paths
  \draw [endnodestyle={rotate=-90},endnode=foo] (6,0) to[bend right] ++(\Ang:2cm);
}
\end{tikzpicture}
\end{document}

Better output, less convenient to use

You could save two coordinates on the path, one very close to the end (I randomly used pos=0.97 here, but I suspect the ideal value will depend on the path), one at the end (pos=1). Then in a separate path, calculate the angle between those two coordinates with the let operation (manual section 14.15), and use that to set the anchor of a node. This does not have the problem the decorations.markings version above suffered, of failing at larger bend angles, but is as mentioned less convenient to use.

\documentclass[border = 2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings, calc}
\newcommand{\addnode}[2][tmp]{%
\path let
  \p1=($(#1-A)-(#1-B)$),\n1={atan2(\y1,\x1)}
  in
  node[anchor=\n1] at (#1-B) {#2};
}

\begin{document}
\begin{tikzpicture}
\foreach \Ang in {0,45,...,350}
{
  % works fine for straight lines
  \draw (0,0) -- 
        coordinate[pos=0.97] (tmp-A) % near end
        coordinate[pos=1]    (tmp-B) % at end
  (\Ang:2cm); % close the path
  \addnode{a} % add node in separate step


  \draw (6,0) to[bend right=80]
       coordinate[pos=0.97] (tmp-A)
       coordinate[pos=1]    (tmp-B)
    ++(\Ang:2cm);
  \addnode{foo}
}
\end{tikzpicture}
\end{document}

(this is the first time I add another answer --- if it's not the correct thing to do, please tell me; but this is so different from my first one that... should I mark it community wiki?)

You can use a transform on a decoration, thanks to the fact that the coordinate system in a decoration is along the path; that will shift the node after the end of the path by a distance specified as the second argument:

\documentclass[border = 2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}[
    endnode/.style n args={2}{
        decoration={
        markings,
        % transform={xshift=0.1*\pgfdecoratedpathlength}, % relative to lenght
        transform={xshift=#2},                            % absolute 
        % maybe add a endnodestyle like in Torbjørn T.'s answer
        mark=at position 1 with {\node[draw=red] {#1};}   % just to debug 
        },
     postaction={decorate},
  },
]
\foreach \Ang/\Len in {0/1,45/1.2,90/1.4,135/1.6,180/1.8,225/2,270/2.2,315/2.4}
{
    \draw [endnode={A}{2mm}] (0,0) -- (\Ang:\Len);
    \draw [endnode={B}{4mm}] (6,0) to[bend right]  ++(\Ang:\Len);
}
\end{tikzpicture}
\end{document}

This makes the node centered at a fixed distance from the end of the path, along the path. It doesn't change the node anchors, unfortunately... and auto is not working here.

Thanks to: Length of curve in TikZ, Torbjørn T.'s answer, \tikzset key with multiple arguments.