You can even carry things other than your children in a trailer. I often see a cyclist in a nearby park cycling with his Golden Retriever in tow. Trailers are also good for bringing home packages from the store when you'd rather take your bike than drive a car. Some disadvantages of trailers are the rough ride. Your child will get bumped around a lot.
Some parents feel less in control with a trailer than with a child carrier.
And, a big disadvantage is price. I chose a child carrier seat for my child. They come in several varieties. Most attach to a rear rack. If you have a mountain bike or touring bike with eyelets, you should have no trouble putting a rack on the back of the bike.