I'm from Europe where there are a lot of low cost airlines like easyJet, Ryanair, etc... But these operate mainly in Europe.
Then, my question is if there are airlines which provide cheap flights within the US.
There is one airline, Spirit Airlines, which is trying to copy Ryanair with super cheap tickets and aggressively bad service. They serve a limited number of destinations, though, so they may or may not fly where you're going.
There are several airlines in the US which were started as low cost carriers. The biggest are jetBlue and Southwest. However, the entire airline business is so competitive that those airlines don't always have the best prices. Every American airline tries to charge $1 less than the lowest competitor, so sometimes you will see amazingly high prices even on "low cost" airlines if the route is competitive enough, and sometimes big airlines will match Southwest's fares.
Your best bet is always going to be to do a search online to find the lowest fare for a given route. Southwest Airlines and Spirit Airlines will not appear on any website but their own, so if you are looking for the lowest fare, you will have to search on their two sites individually.
I always thought that the concept of low-cost airline, or no-frill airlines originates from the US.
According to Wikipedia, the following low-cost airlines can be found in the US. [updated 22/May/2017]:
"Ultra" low-cost (no-frills) airlines:
Another LCC, Virgin America, was acquired by full-service carrier Alaska Airlines in 2016 and is being integrated into the latter's network.
There are a few airlines which operate flights at lower fares. I don't know if it is the same "low-cost" concept as known in Europe. Some names that I remember are
I personally experienced Virgin America. Tickets have to be purchased online and it's $15 for each bag checked as hold baggage. As opposite to Ryanair, most of the aforementioned airlines operate on the same airports as regular airlines and use terminals with the same amenities. One exception is that, in the Los Angeles area, jetBlue lands in Long Beach (LGB) instead of the main Los Angeles airport (LAX).