There are many Eurovision songs that you may wish to sing at your party.  It is almost impossible to find Eurovision Karaoke Videos, so we suggest the best thing to do is to make a CD with the songs on, and print off the lyrics.  Alternatively, you could compile a video of the songs live at Eurovision, so the performers could also copy the dance moves!
Essential equipment to have is a microphone so that you can hear the performer over the original singer's voice.  A karaoke machine is even better, but not essential.

Bear in mind that if you invite non-Eurovision fans, the number of songs they know well will be extremely limited.  Obviously, the songs known by your guests depend largely on where they are from, for example someone from Sweden is more likely to know Diggi-loo Diggi-ley than a guest from Croatia!
Also you should note that foreign language songs (except English ones) will be harder for people to sing, though you could see this as being more fun, and as you probably will have seen recently, with the success of the siblime Dragostea Din Tei, that everyone can (or thinks they can) sing along with foreign language songs just as loudly and out of tune!