I downloaded the application , on my phone samsung SGH-X636 and run application. I type my login details correctly, clicked login, clicked yes to allow access to a non-trusted thingy, it says "connecting..." and then i get error "DNS resolution failed".

Any suggestions?

It seems like your carrier's DNS is unavailable or it doesn't know about chess.com?

Sure thing!  The Domain Name Server (DNS) has not connection to www.Chess.com in Chile, as far as I know.