TikZ: Intersection of two lines

The reason the code does not work as provided is that there is only one intersection, and so (intersection-2) does not exist. One way to alleviate this kind of issue is to specify total=\t to contain the total number of intersections and the use a foreach to loop through each intersection:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}

\begin{document}
\begin{tikzpicture}[every node/.style={black,above right}]
\draw[name path=line 1] (0,0) -- (2,2);
\draw[name path=line 2] (2,0) -- (0,2);
\fill[red,name intersections={of=line 1 and line 2,total=\t}]
    \foreach \s in {1,...,\t}{(intersection-\s) circle (2pt) node {\footnotesize\s}};
\end{tikzpicture}
\end{document}

Reading the PGF Manual helps ;). See page 54ff, I made this from it:

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}
\usetikzlibrary{calc,intersections,through,backgrounds}

\begin{document}

\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (3,3);
\draw [name path=A--B] (A) -- (B);
\coordinate (C) at (3,0);
\coordinate (D) at (0,1);
\draw [name path=C--D] (C) -- (D);
\path [name intersections={of=A--B and C--D,by=E}];
\node [fill=red,inner sep=1pt,label=-90:$E$] at (E) {};
\end{tikzpicture}

\end{document}

which results in:

An alternative, in the form of tkz-euclide.

\documentclass{article}
\usepackage{tkz-euclide}
\begin{document}
\begin{tikzpicture}
  \tkzDefPoint(0,0){A}  \tkzDefPoint(2,2){B}
  \tkzDefPoint(0,2){C}  \tkzDefPoint(2,0){D}
  \tkzDrawSegments(A,B C,D)
  \tkzInterLL(A,B)(C,D) \tkzGetPoint{E}
  \tkzDrawPoints(E) \tkzLabelPoints[below](E)
\end{tikzpicture}
\end{document}

The first three lines define the points and draw the line segments between them. \tkzInterLL compute the intersection of the lines A--B and C--D, while \tkzGetPoint{E} gives the point a name. Finally the point is drawn and labeled.

You can mix this with "normal" TikZ code if you want to, e.g. (borrowing from Tom Bombadil):

\documentclass{article}
\usepackage{tkz-euclide}
\begin{document}
\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (3,3);
\coordinate (C) at (3,0);
\coordinate (D) at (0,1);
\draw (A) -- (B);
\draw (C) -- (D);
  \tkzInterLL(A,B)(C,D) \tkzGetPoint{E}
\node [fill=red,inner sep=1pt,label=-90:$E$] at (E) {};
\end{tikzpicture}
\end{document}

This only uses tkz-euclide to find and name the intersection.