\begin{framed} with background-color

The package documentation is your friend.

\documentclass{article}
\usepackage{xcolor}
\usepackage{framed}
\definecolor{shadecolor}{rgb}{1,0,0}
\begin{document}
\begin{shaded*}
    Let me have a background :(
\end{shaded*}
\end{document}

Using the mdframed package defining the background colour is pretty easy:

\documentclass{article}
\usepackage{xcolor}
\usepackage{mdframed}

\begin{document}

    test

    \begin{mdframed}[backgroundcolor=blue!20] 
        In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs.
    \end{mdframed}

\end{document}

Two solutions with the framed package with a frameborder and a framecolor: I define the frshaded and frshaded* environments. Rule thickness and distance between text and frame can be easily changed.

    \documentclass [a4paper,12pt]{article}
    \usepackage[utf8]{inputenc}
    \usepackage[T1]{fontenc}
    \usepackage{lmodern}
    \usepackage[x11names]{xcolor}
    \usepackage{framed}
    \colorlet{shadecolor}{LavenderBlush2}
    \colorlet{framecolor}{Red1}
    \usepackage{lipsum}

    \newenvironment{frshaded}{%
    \def\FrameCommand{\fboxrule=\FrameRule\fboxsep=\FrameSep \fcolorbox{framecolor}{shadecolor}}%
    \MakeFramed {\FrameRestore}}%
    {\endMakeFramed}

    \newenvironment{frshaded*}{%
    \def\FrameCommand{\fboxrule=\FrameRule\fboxsep=\FrameSep \fcolorbox{framecolor}{shadecolor}}%
    \MakeFramed {\advance\hsize-\width \FrameRestore}}%
    {\endMakeFramed}

    \begin{document}

    Text text text text text text text text text text text text text text text text text text text text. text text text text text

    \begin{frshaded}
    \lipsum[1-2]
    \end{frshaded}
    \colorlet{framecolor}{VioletRed4}
    \setlength\FrameRule{1.5pt}
    \begin{frshaded*}
    \lipsum[1-2]
    \end{frshaded*}

    \end{document}