While you aren't going to get as much variety when it comes to characters on the cruise ships (there's some licensing issues that don't allow some to not appear outside the parks like Winnie the Pooh and friends), you can almost always find the classic favorites like the Fab 5 and a lot of the Disney princesses. And if you happen to be sailing during a holiday or other special event, a few surprises or two!
Most meet & greets occur just like they do in the parks, a schedule is given out each day in your stateroom that will tell you which characters are meeting, at what time(s), and where. All you have to do is show up with your autograph book and camera and be prepared for a hug or two (don't worry if you forget your camera, Disney has their photographers on hand as well with Mickey's camera).
That being said, there are a few exceptions. On certainly sailings there is what's known at the "Princess Grand Gathering" where 4 or 5 of the Disney princesses meet at once. And recently Anna and Elsa from "Frozen" are making appearances as well. Because all of these ladies are so popular (Anna and Elsa have been commanding 4-6 hour wait times in the Magic Kingdom!), a ticket is required to attend their appearances. Tickets are free, and must be picked up before hand (inquire at Guest Services or in the printed schedule on the first day of your cruise for pickup locations). The tickets just give you a time to arrive so that there isn't a mad rush and you don't spend all day waiting in line (similar to FastPass).
Another tip, be prepared from unscheduled character appearances. There were several evenings where we'd stumble across Peter Pan or Aladdin (to name a few) in a corner just waving and chatting with guests, without a handler or Disney photographer in sight. If you are ever curious if you or your kid's favorite character is on board, just inquire at guest services.