box around theorem statement

Here's a minimum working example (MWE) of how one can use the ntheorem and framed packages to draw a rectangular frame around a theorem environment:

documentclass{article}
usepackage{framed} % or, "mdframed"
usepackage[framed]{ntheorem}
newframedtheorem{frm-thm}{Theorem}
begin{document}
begin{frm-thm}[Pythagoras]
Let $a$, $b$, and $c$ denote the lengths of the sides of a emph{right 
triangle}, i.e., of a triangle with one angle equal to $90^circ$. 
Without loss of generality assume that $ale b<c$. Then
[ a^2+b^2=c^2. ]
end{frm-thm}
end{document}

Check the user guides of the framed and mdframed packages for available options for setting the style of the frame.

Here's an example of how you can put some colour into your Theorems using ntheorem combined with PSTricks.

documentclass{article}

usepackage{pstricks}
usepackage{pst-grad}
usepackage{framed}
usepackage[framed]{ntheorem} 
usepackage{lipsum}

% framed theorem, red background
theoremstyle{plain}
theoremsymbol{}
theoremseparator{}
deftheoremframecommand{%
      psshadowbox[fillstyle=solid,fillcolor=red,linecolor=black]}
newshadedtheorem{mytheorem}{Special Theorem}

% framed theorem, gradient shading
theoremstyle{plain}
theoremsymbol{}
theoremseparator{}
deftheoremframecommand{%
      psshadowbox[fillstyle=gradient,gradbegin=red,gradend=yellow,linecolor=black]}
newshadedtheorem{myfancytheorem}{Sunset special theorem}

begin{document}

begin{mytheorem}
lipsum[1] 
end{mytheorem}

begin{myfancytheorem}
 lipsum[1]
end{myfancytheorem}

end{document}

Either run the above code with

latex myfile.tex
dvips myfile.dvi
ps2pdf myfile.ps

or

xelatex myfile.tex

If you want to run it with pdflatex then add the option pdf to the pstricks load

usepackage[pdf]{pstricks}

and compile with

pdflatex -shell-escape myfile.tex

I don't know if the tcolorbox package was available at the time that the question was asked, but here's a small example taken directly from the documentation:

% arara: pdflatex
documentclass{article}

usepackage{tcolorbox}
tcbuselibrary{theorems}

newtcbtheorem[number within=section]{mytheo}{My Theorem}%
{colback=green!5,colframe=green!35!black,fonttitle=bfseries}{th}

begin{document}
begin{mytheo}{This is my title}{theoexample}
  This is the text of the theorem. The counter is automatically assigned and,
  in this example, prefixed with the section number. This theorem is numbered with
  ref{th:theoexample} and is given on page pageref{th:theoexample}.
end{mytheo}

end{document}

If you are using thmtools, it should be noted that it can render boxed theorems per se. Example:

documentclass{report}
usepackage{thmtools}
usepackage{lipsum}

declaretheorem[
  shaded={rulecolor=black, rulewidth=1pt, bgcolor=magenta},
  name=Theorem,
]{thmboxed}

begin{document}

begin{thmboxed}
lipsum[1]
end{thmboxed}

end{document}

renders as

I'd like to use a box around definitions and theorems, but leaving the header outside of it like:

Theorem 3.15: Sufficient conditions for the existance of...
|---------------------------------------------------------------------------------------------------------|
|     Let      .....                                                                                               |
|                 .....                                                                                              |
|                 .....                                                                                              |
|      Then  ......                                                                                             |
|                 ......                                                                                             |
|                                                                                                                    |
|________________________________________________________________|

It's kinda like I have to put the box INSIDE the theorem body, leaving eventually some space for the title and author...