Lets begin by saying that ceramics, pottery, and porcelain are all referenced a bit differently, yet they are all made of clay. The most important thing about this ability to repair ceramics is based around it being clay. To better understand this let me say that clay is structured with tons of tiny particles all attaching together like Legos. Now when a Lego creation is broken, they usually break in a clean manner so that you can put them back together. That clay is the same way.
As you can see in this image, clay can be fit back together so well that you can repair it with gold in the kiln. This way of repair is difficult and not as strong as what I am going to suggest. The convenient and easy route is to use a super glue or epoxy. Now days super glues and epoxies are so strong and waterproof that almost any kind will work.
Once you have your glue or epoxy, follow it's instructions on the label prior to application. Have some rubber bands of something to wrap around your piece so that it will continue to apply pressure after you. When your ready just use a tiny bit of adhesive in the center of where you gluing to try and avoid any from leaking out the sides. After the glue is on, attach the two broken pieces and apply a generous amount of pressure for some odd amount of time. Right after that, wrap your rubber band/s or what not around the piece.
Follow the instructions on the label when it comes how long to dry the glue. Once you are done your piece will be back for more. Depending on how bad the breaks are, they may show more or less of a crack after the repair. If your piece is for liquids, food, or microwave use, I recommend not using it any more or just replace it and not do this work. It is super easy to repair clay thanks to how it fits right back together. Your first repair might not be the best but the more you do it, the better and more comfortable you will get doing this.