Note here that while this answer covers BMI and ASCAP, the general process is the same with other songwriter royalty groups, like PRS.
The easiest way to join ASCAP or BMI is by visiting their respective websites. The entire application process can be handled online - this is true for both songwriters who want to join and for publishers who want to join. Each group has a one-off fee involved in applying.
What they are looking for in an application is someone whose songs have a realistic chance of being played in the media or in a public setting at some time in the near future. If you have just recorded a demo, but you haven't done anything with it yet, then it's probably not the best time to file an application. However, if you have a website and your songs are online, then your application will likely be accepted. If your application is denied for some reason, take heart. It's not an indictment on you or your music; it just means you're not quite ready to join yet. As your career progresses, you'll get a thumbs up from the groups.
Note that BMI and ASCAP are songwriting royalty groups, and so they are only for the songwriter in the group. Musicians need not apply here. Also, note that you cannot apply to BMI if you are already an ASCAP member.
Choosing a Royalty Group
What if neither BMI or ASCAP works for you? Then you can consider membership with SESAC. SESAC performs the same function as BMI and ASCAP, but with one significant difference - SESAC does not accept every applicant. While the approval process for BMI and ASCAP is largely a formality to ensure you meet very broad criteria, SESAC membership is by invitation only. Each member is vetted to ensure that they fall into the group's framework.
Does that mean that SESAC membership is the ultimate prize you should aspire to? Not necessarily. Each group performs essentially the same services for their members, and there is little reason that belonging to BMI or ASCAP can't be equally successful for you - after all, they represent thousands of big names. There may be an argument to be made that the exclusivity of SESAC membership has its benefits and that having a smaller, curated stable of artists makes for more personalized service. However, these things shouldn't deter you from being perfectly happy seeking membership in either BMI or ASCAP.
When you're choosing between BMI and ASCAP, there are several other factors to consider. Some people make the decision based on who their favorite musicians have chosen to work with. Others take a cue from the artists in their genre. Still, others are swayed by the history of BMI vs. ASCAP (payola, race records, and more). For most songwriters, the differences are negligible, unless you are offered a publishing deal that is somehow linked to one of these groups.
The Role of Musicians
Musicians often wonder what their roles are in BMI and ASCAP deals, particularly if the songwriter is also in the band. In reality, their role is nothing. These royalties are reserved for the songwriter and publisher, and so if you only play on a recording, but didn't write it, membership is not for you. However, if you are given credit on a song - for instance, the main songwriter agrees that you have written 10 percent of a track - then you can claim your membership so that you are paid your appropriate percentage as a songwriter on the track.
If there is any confusion in your band about who writes the music or how song ownership is split, the time to talk about it is now - especially before anyone joins one of these groups. Because the songwriter stands to make significantly more money than the performers, this issue can be very contentious. It is best to be clear at the outset - or have a conversation as soon as possible - about songwriting to avoid future conflicts. You may be surprised how differently two people can view the same songwriting process and disagree on the contributions made by each. Negotiate these terms before anyone registers a song to prevent confusion.