International Flights to Panama
Panama, or more specifically its capital Panama City, is the hub of travel in Latin America, with most regional (as in Latin American) airlines flying there.
From North America, Panama City is served by American Airlines, Delta, Continental and Copa.
The only European airline with direct flights to Panama City is Iberia (from Madrid). However, the cheapest tickets are usually found with Air France or KLM, who will route you via a North American city with a connecting flight with either Copa or Taca. Lufthansa also fly to Panama City, via Caracas, as well as British Airways (via USA), which is normally rather expensive.
Copa
Copa is Latin America's leading airline, offering a huge network of flights. If you're travelling from somewhere other than North or South America, remember than Copa flights are almost always much cheapest when booked as a Hahn airpass (also known as the All America Airpass - see our blog for more advice).

Copa Route Map
Panama Domestic Flights
Two airlines offer domestic flights in Panama - Aeroperlas and Air Panama. Both airlines fly to the most popular travel destinations such as Contadora (Pearl Islands), San Blas Islands and Bocas del Toro. Generally speaking, Aeroperlas has a more comprehensive selection of flights to Eastern Panama, while Air Panama flies to more destinations in West Panama. Air Panama also has flights to San Jose in Costa Rica. Nature Air also offers flights from San Jose to Bocas del Toro.

Aeroperlas Route Map

Air Panama Route Map - note that not all of these destinations are served (eg. Coiba Island!), so why Air Panama promotes them on their route map is beyond me...