In what episode do Ash and Dawn do the handshake?

I'm guessing the Ash and Dawn you are referring to are from Pokemon?

There is not a handshake between the two as such, but they do high-five in episode 659* "Memories are Made of Bliss!", which takes place in season 13, Diamond and Pearl Sinnoh League Victors.

The episode was aired in Japan on 9th September 2010 and in the US on 5th February 2011.

The high-five takes place as Ash, Dawn and Brock say their goodbyes. Brock has decided to leave the journey and the group come to the conclusion that they all have different things that they need to do. As they stand at the boat Dawn raises her hand to invite Ash to high-five, which he does.

*There seems to be some dispute online as to the episode number, with different sources listing it as 657, 659 and 660.