Katie the kitten has appeared in previous Animal Crossing titles. In previous games, she would randomly appear in your town, crying because she got separated from her mother and needs to be escorted to where her mom is. In New Leaf, Katie is more confident and goes traveling on her own. She'll ask you to escort her to a new town.
Katie appears in your town if she was escorted by one of your friends to your town three days ago. She will be in your town between 6AM and 6AM the next day, or until you take her to another town. It is also possible that Katie may appear in your town randomly if you have ever connected to the Internet in your New Leaf town.
When Katie is in your town, talk to her and she will ask you to take her to another town. Any town will do. Ask one of your real-world friends to open his or her gate so you can travel to that town. Katie will follow you. When you get on the train, Katie will either come with you, or she will drop her ticket and get off the train, but the doors will close behind her and you'll go to your friend's town by yourself. When you go back to your own town, Katie will not be there.
After you have gotten on the train with (or without) Katie and she isn't in your town anymore, you will get a letter from her the next day. If she dropped her ticket, her letter will explain that she got on the next train. Attached to her letter will be an exclusive item that you can only get from Katie.