Grout is the material which goes in between the tesserae (the pieces which make up the mosaic). You can use ready-mixed household tile grout or a cement-based grout. This is more waterproof. It can be bought as a powder, to be mixed with water, or you can make your own - the most simple recipe is one part cement to three parts sand. Cement is very caustic so you need to wear gloves and take safety precautions.
Water is added to a small heap of powdered grout and mixed to a creamy consistency, then spread it on, forcing it down into all the gaps with your fingers or a spreader.
As the grout dries on the surface of the tesserae, brush it off with an old toothbrush. Work inwards from the edge, taking care not to dislodge grout from between the pieces of crockery. Cleaning off grout at this stage will save a lot of work later, although set grout can be cleaned off with sandpaper. Allow the grout to set overnight, but don't put it in a warm place as slow drying will give the piece more strength. Go over it again with a brush and then a soft cloth to remove the remaining haze and dust of grout.
Project 2 gives you an idea of how different grout colour can affect the look of a mosaic.
Whether you use a ready-mixed grout or a cement-based powder grout, you can change the colour of it by using acrylic paints. Bear in mind that this may change the consistency (and perhaps the drying time) of your grout, and it takes an awful lot of paint to affect cement grout if it is already grey or brown. There are, however, dyes available for colouring cement. You can even paint acrylic paint on to the grout when it's dry, although this can be very fiddly.
If you're colouring the grout, mix up enough to do the whole of your mosaic. Otherwise it's very difficult to match the colour in a second batch. Remember that the grout will be a bit lighter in colour when it has dried.
A couple more tips: if you can't decide on a grout colour for a particular piece, make a small sample using the same materials, perhaps on a scrap of board or the back of a tile, and try grouting different sections with different grouts to see how they work. I have done this even with grey grout, to decide how dark I would like it, gradually adding grey grout to white, or black to grey.
Also, experiment with just using one colour of shards and try different grouts all over or on different parts of the mosaic. You can get some good results using bright coloured grout with just plain white crockery. A dense colour may stain any scratches of crazing in the glaze, which can be an interesting effect.