A booking agent (also known as a promoter) is responsible for organising gigs (DJ sets, live performances, concerts, etc.) and negotiating the best deal for you. A booking agent often works for an agency which has its own network of venues (clubs, halls, festivals, etc.) of varying sizes that they can place you in. More precisely, that they can sell you in. Or rather, your unique talent!
You are Publisher
You want to Manage your author's rights
Your creations are your intellectual property and they generate authors' rights with fair remuneration for their use. However, in order to benefit from this revenue you need to register your work, and all the necessary filing and reporting procedures must be followed.
Success! Your music is starting to become known! You're getting more and more bookings and organisers are coming to you. But you don't have the time to spend sending email reminders and doing admin work. You need a booking agent!
As well as finding gigs, a booking agent often takes care of the logistics of your bookings: transport, accommodation, subsistence, etc., either by negotiation with the venue where you're playing, or paying for it out of the fee he/she has negotiated for you.
The booking agent also takes a commission, which is their only remuneration for the administrative work they handle on your behalf (contracts, tax returns, payments, etc.).
Despite all these deductions from your fee, there are definite advantages to having a booking agent: they increase the number of gigs you get through their professional network, and help you develop a strategy for live performance; for example, concentrating on one particular music scene or certain periods of the year, which is more practical if you live abroad and are gradually establishing yourself.
It depends very much on the relationship you have. Ideally, you must establish and maintain a relationship of trust; many practical aspects of your life as an artist depend on it!
Just like finding a record label, the best thing to do is involve yourself in the music scene that you're interested in and make contact with the people involved in it. You might meet the right person at a festival or club you often go to; try to overcome any awkwardness you feel, and generate contacts. Talk about your music and get it played. You could have some business cards printed with links to your SoundCloud, MixCloud or Facebook page printed on it, for example.
You can also do a search on social networks. Artists' Facebook and SoundCloud pages often list contact details for their booking agents or promoters - try them! Don't get too ahead of yourself though - don't send your first song off to somebody who manages the likes of David Guetta or Seth Troxler! Try to find a manager who looks after artists similar in outlook to yourself.
And above all, obviously the hardest thing to do - perform, as often as you can!
Performing is valuable experience and is the best way to get discovered.