The skin library of tcolorbox has the feature of attached box titles, which can be styled on their own, as well as being shifted with xshift and yshift options.
To attach the box title top right, just say attach box title to top right={...}, with some options, see the manual (version 3.70, as of now, page 131 for more options and possibilities)
\documentclass{article}
\usepackage[most]{tcolorbox}
\newtcbtheorem{Summary}{\bfseries Summary}{enhanced,drop shadow={black!50!white},
coltitle=black,
top=0.3in,
attach boxed title to top right=
{xshift=0em,yshift=-\tcboxedtitleheight/2},
boxed title style={size=small,colback=pink}
}{summary}
\newtcolorbox[auto counter]{summary}[1][]{title={\bfseries Summary~\thetcbcounter},enhanced,drop shadow={black!50!white},
coltitle=black,
top=0.3in,
attach boxed title to top right=
{xshift=0pt},
boxed title style={size=small,colback=pink},#1}
\begin{document}
\begin{Summary}{}{firstsummary}
Brontosaurs are thin at one end, thick in the middle and thin again on the other end
\end{Summary}
\begin{summary}[boxed title style={colback=yellow},colback=green]
Brontosaurs are thin at one end, thick in the middle and thin again on the other end
See excavations of Brontosaurs skeletons
\end{summary}
\end{document}

Here, two versions are shown, own with an overlay of the box title and own with almost no boundary space between bottom of the title box and the top of the regular tcolorbox frame. In my point of view, the first version does provide the better look.
\documentclass{...}and ending with\end{document}. – Jul 24 '15 at 07:08