There are three places where music can be played in relation to your vows... before, during, and after.
Before Your Vows
An instrumental, vocal, or solo before your vows can help ready your guests for the moment to come. It can also be used as a breather for the Bridal Couple, and a chance for them to prepare for the upcoming words they will be sharing. This is especially helpful if you have decided to recite your vows from memory, or even if you'll be reading them.
A song just prior to the vows will give you a chance to go over the vows in your mind, or on paper. Even if you'll only be repeating after the Celebrant, a song before your vows can give you a chance to catch your breath, take in the moment, and mentally and emotionally get ready for the highlight of the ceremony.
Also, having a soloist perform at this time will tend to shift some of the focus from you onto the soloist, so any last minute preparations you must make for the vow recital will be less obvious. (e.g. having the maid of honor hand you your vows, or perhaps [and we sincerely hope not] having to locate your vows if you've misplaced them!).
During Your Vows
You can have music played during your vows. The same rules apply here as to the background music for readers. You want the music to be a softly played instrumental (so you're not competing with it to be heard by your guests), you want it to fit the words of your vows (e.g. an instrumental version of "From This Moment On" might be appropriate), and you want it to fit the theme and/or pre-planned mood of your wedding (if your wedding has either).
After Your Vows
Here you could use an instrumental, vocal, or solo piece. If using an instrumental, the music should be more reserved in nature, giving yourselves and your guests a chance to reflect or meditate on the moment that has just taken place. A vocal piece should probably be one pertaining to marriage or at least great love. For example, "The Wedding Song" by Paul Stookey or again "From This Moment On" by Shania Twain & Bryan White.
It might even be a nice touch to have a soloist sing a song to the both of you, once again with a wedding or romantic theme. Some couples have actually used the words of a particular song as part of their vows, and followed their vows with that very song! That can be very dramatic (not to mention romantic) indeed!
The Ring Exchange
All of the above mentioned conditions for the vows can be applied if the vows and ring exchange are being performed together, or even if the ring exchange is a completely separate activity. Some couples have chosen to recite their vows without music, while performing their ring exchange with music, and vice versa. A couple of good vocal songs for the ring exchange are "With This Ring" by Sawyer Brown, or a different song of the same name, "With This Ring" by Kenny Loggins.
Congratulations! You've just been pronounced a married couple! Your going to need some music for that trip back up the aisle. Check out some great recessional songs here.