Just for the record, I am totally pro kids at weddings. That being said I can understand why someone would choose not to invite children to their weddings.
Kids are fun to have a weddings because they are adorable all dressed up and they run around and entertain everyone. They are not so fun when they are screaming and crying during the vows, or destroying decorations it took you 3 months to make.
The real problem with kids are that they get bored, and tired and that's when they start to fall apart. Keeping them entertained is the key to having a successful wedding with kids.
The Guest List
If you have lot of guests traveling to the wedding, you may what to consider inviting kids. Many guests may not want to leave their kids for an entire weekend (many may jump at the chance!) It may also cause hurt feelings if the couples nieces and nephews are attending and they were not allowed to bring their own children.
I am all for having a babysitter at the church for the ceremony. Alot of times, if it is a church you attend regularly, you could probably ask one of the regular nursery monitors to watch kids during the ceremony, please pay them. If they are unavailable, ask one of your friends that has children to recommend a sitter (or sitters) that they trust to monitor the children, so their parents can enjoy the ceremony. If you are not having ceremony in church, ask you venue contact if they have a room that could be set for the children, fill it with coloring books, crayons, and maybe a portable DVD player.
Children in the Wedding
Oh, the little scene stealers. These are your flower girls, your ringbearers. They are often super cute and you love them a ton. That being said, you don't need them standing between you and your soon to be spouse while you say your vows. I always suggest that flower girls and ring bearers walk down the aisle and then move to sit with their parents (if one of them is not in the wedding), Grandparents, or a designated friend that could remove these cutie pies at the first sign of restlessness.
Children should sit with their parents. If both parents are sitting at the Bridal party table, they should sit with the Grandparents or the family friend that is 'in charge' of the child. If you have the room, and an extra table, I would suggest creating a craft table for the kids, someplace where they can color or make a picture, maybe a card for the bride and groom. I would position it near the photo booth, and you never know maybe the adults might get alittle crafty!