Generally I'll buy anything that seems cool for a reasonable price, but these days I'm reluctant to buy new games because my backlog is so enormous it would take a lifetime to play all the games I already have to completion. Not that I would want to finish every game, but it still comes down to "I could buy this new thing, or I could play the 5 games I bought in the last Steam sale that I still haven't touched". I still make an exception if something is especially cool though, or if I just want to support the developer.
As for convincing me a game is worth checking out: basically, have a great gameplay trailer... no cinematics, no endless logos and backstory... show me the game. Word of mouth helps, certainly, but the game needs to look interesting.