First you need to brew some beer, and bottle it. But not any bottle will do, you need brown or green glass between 10 and 14 ounces. No swing-tops. Next you need to fill out a Bottle ID Form. The AHA website has links to this sucky form, and this sucky form for Club-only competitions, but what you really want is this good form.
What makes the good form good? It's a PDF with fillable fields, so you can type in your name, address and other info rather than writing it. But the best part is that when you fill out the first label (there are four per sheet) it fills in the other three labels auto-magically. The sucky forms are just pictures of a form that you have to print and fill out by hand for each bottle.
If you are entering a competition that only requires two bottles and you want to save paper you can fill out most of the form, print it and fill in the beer name and category by hand. Cut the label sheet into four parts, and rubber-band one to each of the bottled beers you are entering, and you're almost done.
If you need to include a recipe with your entry, the AHA website has links to this sucky one, this sucky one for Club-only competitions, and this good one. The links to the good forms are strangely hard to find. I used the good ones for a Club-only competition last year and the contest organizers were fine with it.
The final thing is to write the check and drop off the bottles, and you're all set!