That’s a very good question.
Now that Ryanair has partnered with Air Europa to offer flights from North America, your options for cheap travel to Europe have increased.
Ryanair is perhaps the best known of all the ultra low costs carriers (ULCC). They broke a lot of new ground when they started, offering cheap flights which require you to pay extra for just about everything. Things like paying extra (often more than the actual fare) for checked luggage, seat selection, etc. Many of the larger airlines are now offering similar basic fares. Since flying times in Europe are relatively short, if you don’t mind traveling with a carry-on and you don’t care where you sit, you can travel very cheaply by air in Europe with a bearable amount of discomfort.
This article in Travel+Leisure is very informative. If you don’t live in the European Union, pay special attention to the last paragraph in the piece, or you may miss your flight.