How to draw a Venn diagram in TikZ?

documentclass[pstricks,border=12pt]{standalone}
begin{document}
begin{pspicture}(8,6)
    psframe(8,6)
    psset{fillstyle=solid,linecolor=cyan}
    pscircle[fillcolor=cyan!50](5.25,3){2}
    pscircle[fillcolor=white](2.75,3){2}
    pscircle[fillstyle=none](5.25,3){2}
    uput{5pt}[45](0,0){$emptyset$}
    rput[r](2.75,3){Pat=9}
    rput[l](5.25,3){Env=242}
    rput[b](4,5){$B-A$}
    uput{2pt}[45](8,6){$H$}
end{pspicture}
end{document}

I will go about it differently. Instead, I will draw the circle as nodes. Then, with the fit library, draw the borders. This way, you can write the labels relative to the named nodes.

The Code

%https://tex.stackexchange.com/questions/101839/tikz-venn-diagramm
documentclass[tikz,border=5]{standalone}

usetikzlibrary{positioning,fit,calc}

begin{document}
begin{tikzpicture}
colorlet{circle edge}{blue!50}
colorlet{circle area}{blue!20}
tikzset{
     filled/.style={fill=circle area, thick,inner sep=0pt}, 
     outline/.style={draw=circle edge, thick,inner sep=0pt}}

% The circles
node (secondcircle) [circle,filled,text width=3cm] {};
node (firstcircle) [circle,left=-1cm of secondcircle,outline,text width=3cm, fill=white] {};
draw [outline] (secondcircle) circle (1.5cm);

% The labels
node at ([xshift=-0.15cm]firstcircle) {$mathrm{Pat=9}$};
node at ([xshift=0.4cm]secondcircle) {$mathrm{Env.=242}$};
node at ($(firstcircle)!0.5!(secondcircle)$) {180};

% The rectangle and labels

node (box) [fit=(firstcircle)(secondcircle), inner sep=1cm,draw,rounded corners] {};

node at (box.north) [anchor=north] {$B-A$};
node at (box.south west) [anchor=south west] {$emptyset$};
node at (box.south) [anchor=north] {432 total};
node at (box.north east) [anchor=south west] {$H$};
end{tikzpicture}
end{document}

The Output

BTW, if you have a lot of Venn Diagrams of two or three sets to draw, I advise you to try the package venndiagram by Nicola Talbot.

Update

I was not so sure at first what should be placed where, but the code below seems to be a more complete attempt.

documentclass[tikz,border=5,convert={density=150}]{standalone}

usetikzlibrary{positioning,fit,calc}

begin{document}
begin{tikzpicture}
colorlet{circle edge}{blue!50}
colorlet{circle area}{blue!20}
tikzset{filled/.style={fill=circle area, thick,inner sep=0pt}, outline/.style={draw=circle edge, thick,inner sep=0pt}}
% The circles
node (secondcircle) [circle,filled,text width=3cm] {};
node (firstcircle) [circle,left=-1cm of secondcircle,outline,text width=3cm, fill=white] {};
draw [outline] (secondcircle) circle (1.5cm);
% The labels
node at ([xshift=-0.8cm]firstcircle.north) [anchor=south] {$mathrm{Pat=422}$};
node at ([xshift=0.8cm]secondcircle.north) [anchor=south] {$mathrm{Env.=189}$};
node at ($(firstcircle)!0.5!(secondcircle)$) {180};
node at (firstcircle) {9};
node at (secondcircle) {242};
% The rectangle and labels

node (box) [fit=(firstcircle)(secondcircle), inner sep=1cm,draw, ultra thick,rounded corners] {};

node [below=12pt of box.north] {$B-A$};
node at (box.south west) [anchor=south west] {$emptyset$};
node at (box.south) [anchor=north] {431 total};
node at (box.north east) [anchor=south west] {$H$};

end{tikzpicture}
end{document}

The coordinates with respect to the named nodes are called anchors. center is the only anchor present in all shapes. So if you name an anchor as A, then you can position another object, say, at A.center. Some more anchors are north, south, east, west, etc. You can see more of these in the pgfmanual. Just type and enter texdoc pgf in your terminal. In version 2.10, you can see some explanations starting from Section 16.5 on Positioning Nodes.

It is common to use ellispes in Venn diagrams. For completion, I show here how draw a Venn diagram with ellipses.

There are many ways of drawing an ellipse in tikz (see here for more details). Below there is a simple suggestion, using the ellipse option in draw.

documentclass{article}
usepackage{tikz}

begin{document}

begin{tikzpicture}

    node at (1,2) {Number of total episodes of case};

    draw  (-2.5,-1.5)  rectangle (4.5,1.5);

    draw[fill=blue!70,fill opacity=0.5] (0,0) ellipse (2cm and 1cm);
    draw[fill=red!70,fill opacity=0.5] (2,0) ellipse (2cm and 1cm);
    
    node at (-1,0) {$9$};
    node at (1,0) {$242$};
    node at (3,0) {$180$};
    node at (-1,1.2) {Pat.${}=180$};
    node at (3,1.2) {Env.${}=422$};
    node at (-2.3,-1.3) {$emptyset$};
    
end{tikzpicture}
end{document}

PS: I added the answer here because this is the most popular post in TeX.SE about Venn diagrams.