Never played in public before? New to playing professionally? Check out these sections: Advice on Becoming a Professional Musician (especially the Essential Equipment section), and Avoiding Performance Anxiety

Probably the two most important things to remember are: Be well prepared, and don't panic!
Nothing, and I mean nothing, can replace good solid planning and preparation. This applies to the music you choose as well. If you need to learn a piece in a short period of time, be sure to pick a short, simple arrangement. If you want to learn a long, elabourate arrangement, give yourself plenty of time. Trying to practice the way some people cram for exams can lead to disaster. If you've only just barely practiced a piece so it's playable, the extra stress of the performance day can be enough to throw you right off kilter. But if you're well practiced, it's a lot easier to deal with the odd mistake or any added confusion (people talking, spontaneous changes in the timing of the ceremony, etc.). Be sure you know exactly when and where you're supposed to play. Find someone (e.g. the minister) to give you cues if necessary, and if there's a ceremony program, be sure to grab a copy. Have a copy of your contract with you, especially if includes any details about the ceremony music. Have a list of what songs you're playing where. Even if you have all the songs and the order memorized, keep the list out anyhow - it's amazing how easy it is to forget what to do where when you're under stress.
Now, I don't want to scare anyone by all this talk of stress - it's just a natural part of playing for any kind of ceremonial occasion. Weddings, especially, are high-stress situations - there's a lot of planning involved, and a lot of things that can go wrong. Normally, the little glitches that can come up are minor and easily dealt with. But you will be spending the entire time alert and watching for cues, so even when you're sitting there not playing, you're still working, if only by being a good observer. There are lots of things you can do to avoid a panic situation, and even if one comes up, to ensure that you can deal with it smoothly.

1. As mentioned above, be prepared - Know your music, know where and when to play, and exactly who you will be getting cues from. It helps to talk to the minister ahead of time. Ushers can also be helpful to talk to, especially if you are supposed to play, say, for the mothers as they are led down to their seats. If you are playing with an organist, find out exactly who's doing what; the organist can also be a good person to get cues from.
2. When preparing a set list for preludes and interludes, make it longer than you think you need. Weddings are nortorious for starting late and going long - especially for more elabourate ceremonies like those for, say, a Catholic wedding with full mass. A lot depends on how long the readings and speeches are, and whether the bridal party themselves are well prepared. Don't worry - many weddings also start and end on time, and usually even if they are late, it is only by five or ten minutes. (Note: If you end up playing significantly longer than you've been hired for, you should get monetary compensation for it - this can usually be accomplished by sending an invoice later, so as not to disrupt the events of the day). However, it never hurts to have several extra songs on hand. Also, be prepared to repeat songs if necessary, and also be prepared to stop unexpectedly. Often the people in the bridal party are nervous, and will walk quickly - it's amazing how fast three bridesmaids can get down the aisle!
3. Find out what kind of mood is being set, and play to that. Some people will want straight traditional, others may want a kind of fairy-tale atmosphere, while others might want casual or upbeat. (I have played everything from very serious traditional church weddings right through to alternative or even wacky ceremonies - one friend even had her processional played on kazoos, no joke!). Be prepared for the unexpected - while you might find most weddings you do are pretty similar to one another, you will probably at some point get, say, a dog as a ring-bearer, or the entire wedding party in Star Trek uniforms. Hopefully, you will be made aware of these kinds of things ahead of time, but occasional surprises do occur.
4. The Processional - This refers to the music that plays the bridal party down the aisle. If it is a small party, sometimes they will only request one piece. However, it is more common to have one piece for the bridal party (any combination of bridesmaids, ring-bearers, flower-girls, etc.) and a separate piece for the bride. If a longer piece is requested (e.g. Pachelbel's Canon in D), you will probably only be able to play a small portion of it, unless the processional is quite long. You can always mark in spots where you can jump ahead or repeat, and sometimes you can get abridged versions of longer pieces; for some pieces, you can just use some sections and leave out other sections all together.
5. Signing the Registry - This always requires music of some kind, since it is a bit of a lengthy process, anywhere from 2-4 minutes long. In this case, long pieces are good. Since four-minute pieces may be hard to come by, you can also string together several pieces in a "set". Having everything in the same key is helpful in this case. You can also repeat a shorter piece two or three times, in which case it helps to have a couple of variations worked in (even if you just change what octave you play in, or the ornamentation). The audience really has nothing to do at this point (it's not really very exciting to watch a bunch of people sign documents), so they will be paying attention to the music (whereas for the processional and recessional, they are usually far more interested in watching the bride and groom and taking pictures). Try not to make the music too sombre. For most weddings (especially those in a church) I usually choose something reflective and/or romantic. Upbeat and lively tunes I usually save for the recessional and afterwards.
6. The Recessional - This is the piece that plays the bridal party back down the aisle and out of the building (or out into the yard/field if outside). Even for serious church weddings, the main sound of this piece is usually "celebratory" - something joyous, whether majestic or upbeat (Celtic pieces are often good for this). As with the processional, this may be a request of a favourite song of the bride and/or groom.
7. Other Interludes - There may be other occasions in the ceremony where music is requested, such as for candle-lighting or during Communion. This will vary depending on the religion, location, desires of the bride and groom, etc. This is where it really comes in handy to have a program, so you know exactly what's happening when. Be sure you have a general idea of how long the interlude is, so you can pick an appropriate length piece, unless they have requested a specific song.
8. Special Requests, Rehearsals, & Accompaniment - If it is requested that you learn a new piece (such as a hymn, or other tune) for the ceremony, be sure you get the music well ahead of time; you may also want to ask for a rehearsal fee if you need to spend any significant time learning it. If asked to accompany a singer or other musician, try and set up at least one rehearsal with them. If that is not possible, try to arrange to meet with them early before the ceremony and practice at least the beginnings, endings, and speed of each song. Make sure you all have the same piece in the same key! If you are asked to attend the wedding rehearsal, don't be afraid to charge for it; the same goes with rehearsing with a singer, or learning a new piece. Normally the fee for rehearsing is less than for playing at the event - for instance, you might decide on a percentage of your regular hourly rate (e.g. anywhere between 10% and 60%, depending on the inconvenience to you), or you may agree on a flat rate after discussing it with your client.