How to draw frame with rounded corners around box

Here's a method that doesn't require loading TikZ explicitly. (Although mdframed uses tikz behind the scenes. This is, however, the package for framing boxes.

\documentclass{article}
\usepackage[framemethod=TikZ]{mdframed}
\usepackage{lipsum}
\begin{document}
\begin{mdframed}[roundcorner=10pt]
  \lipsum[1]
\end{mdframed}
\end{document}

If you don't even want to use TikZ as a backend, you could use PSTricks:

\documentclass{article}
\usepackage[framemethod=PStricks]{mdframed}
\usepackage{lipsum}
\begin{document}
\begin{mdframed}[roundcorner=10pt]
  \lipsum[1]
\end{mdframed}
\end{document}

[Note this doesn't seem to be working for me at the moment.]

Although when seamus wrote his answer, tcolorbox was not in CTAN, it was announced just one month later. And as nobody has written an answer mentioning it, I do it.

So here you have a little code which shows how to declare a framed box with tcolorbox. How to change it's width, frame color, background color, text alignment, space between text and frame border and rounded corners diameter.

For more funny options you can consult tcolorbox documentation or dive here

\documentclass{article}
\usepackage{tcolorbox}
\begin{document}

\begin{tcolorbox}
Here is some text
\end{tcolorbox}

\begin{tcolorbox}[width=5cm]
Here is some text
\end{tcolorbox}

\begin{tcolorbox}[width=.5\textwidth, colframe=red]
Here is some text
\end{tcolorbox}

\begin{tcolorbox}[width=8cm, colframe=red, colback=blue!30, halign=right]
Here is some text
\end{tcolorbox}

\begin{tcolorbox}[width=.5\linewidth, halign=center, colframe=red, colback=blue!30, boxsep=5mm, arc=3mm]
Here is some text
\end{tcolorbox}

\begin{tcolorbox}[width=7cm, colframe=red, colback=blue!30, arc=3mm, sharp corners=east]
Here is some text
\end{tcolorbox}

\end{document}

This another option:

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning,calc}
\begin{document}
    \begin{tikzpicture}[x=0.5mm,y=0.5mm]
    \coordinate (a1);
    \coordinate[right=150 of a1](a2);
    \coordinate[below=90 of a1](a3);
    \coordinate[right=150 of a3](a4);
    %% 
    \draw[ultra thick,rounded corners=10] ($(a4)-(50,20)$) rectangle +(50*2,20*2);%Rectangle rounded
    \end{tikzpicture}
\end{document}