I've entered quilts in county fairs, quilt shop shows, guild shows, and national shows. I've also entered quilting magazine and manufacturer contests.
The most important thing is to follow the instructions to the letter. Make sure you do your best to give the judges no reason to disqualify you or dismiss your work out of hand.
Design and color are important. You need the "Wow factor." It is huge to stand out in a positive way and to be original.
Technique is tremendously important. Everything has to be perfect to win a contest or a big show. The judges tend to be very picky, and if the competition is fierce you don't want to lose a ribbon because you missed a few un-trimmed threads.
I've won a Best of Show, a Judge's Choice, and a First Place in a national show. These were juried shows where you had to submit slides. The photography has to be good to have a chance to even get in.
Once you're accepted it depends on the judges and the competition. The one good thing about juried shows is that you know you have a chance going in.
Contests are a different thing altogether. Not only do you have to be original with perfect technique you also have to keep in mind the reason for the contest. A fabric manufacturer is expecting your submission to help them sell their fabrics. If they are promoting a particular line make sure you use that fabric, even if they don't say you have to. If your piece can make buyers want that fabric, you've got a great chance to win.
It's the same with magazine contests. One thing I would recommend is that if you have to send in your piece, make sure you present it well. I won the Mojo Doll contest for Quilters Home by sending my doll in custom packaging. It made it stand out.
I think competing isn't for everyone because it's such a crap shoot. If you go in expecting to win, odds are you'll be disappointed. However, if you want to test yourself and your skills it's worth doing even if you don't win. I had a lot of disappointments but I kept moving forward.
I don't compete anymore because I prefer to enjoy making quilts instead of focusing on making winners. It's stressful and not as fulfilling as I thought it would be. I thought that getting a Best of Show would change my life, it didn't.