I would like to add that the introduction/application should be somehow professional and clear. I think it matters how you introduce yourself and try to sell your music to organisers. Be polite and humble, send applications to same places also next year. It will pay off some day
Another trick is to ask your friends who have played at a festival to recommend you to the "booker"
If you don't have friends who play - you can go to a festival and make that friendship happening.
getting to festival lineup is not that easy, there are 1000* artists out there , and much less of spots to play.
i know many artists who got bookings just by sending email to festival.
you need to understand what festival needs, and usually they need a combination of two:
A - high quality music that people would enjoy (music must be compatible with concept of event)
B - artist that would attract visitors
So my tips would be to gain both quality and popularity :
release music (ektoplazm, compilations)
promote yourself in internet and your circle of friends
join "notorious" label
play at small parties
play at smaller festivals
Another trick is to investigate who is making the lineup and to talk directly with this person, but if you are not A and B that will not lead anywhere serious.
When i was starting nobody wanted to book me, so my solution was to make a lot of DJ mixes in order to be noticed in internet, and probably more importantly - organize my "small" events in order to get confidence, build fan base and by this - get noticed.
Sounds of Snow reacted to andorra in Tips and tricks on how to apply to play at festivals