Where are the most affordable places to travel in Europe?

That depends on your definition of affordable, almost everywere in Europe you can travel without braking the bank (even to places like Paris), the key is to try to save money on accomodation (which is normally the biggest expense), still if you are looking for new places to visit, these would be my favorites:

• Prague, I know very turistic but is still affordable in many ways since it isn't a euro country yet.
• Krakow (I insist this is one of the best keep secrets in Europe) less crowded than prague and cheap because is not Euro yet, also is a student city therefore you will find better prices on almost everything there.
• Budapest, it feels bigger that Prague but also has the great hospitality of Prague and other Eastern Europe places and is affordable for the same reason above, is not Euro yet.
• Brasov in Romania, the place is reachable with flights on wizzair, is very affordable since is not Euro yet, and the city is just Beutiful and small and the people is amazing.

For more western Europe

• Berlin: Is more a student city so is surprisingly affordable for a place in a developed country such as Germany.
• Lyon: this is one of the best cities in France to travel, is well known for its great food, is affordable, has many places to visit and if you go at the beguinning of December you can catch the Fete de Lumiere which is their main event.

Again my main suggestion is to try to save on accomodation mainly by couchsurfing that way you will not breaking the bank almost anywhere you go.