One way of finding an alien civilisation would be the detection of a large artificial structure, like the ringworld depicted in this artist's impression. Such a structure was first thought of by the science fiction author Larry Niven in his book Ringworld. However, the chances of actually finding such a structure are really low.
ESA/Hubble, M. Kornmesser
CC BY 4.0
<a rel="nofollow" class="external text" href="http://spacetelescope.org/videos/hubblecast97c/">Ringworld (artist’s impression)</a>