Of course, there are many styles and methods for painting fur. But I've had a few people ask me how I paint fur, so here's one way of doing it.
There are a lot of different kinds of fur that require different techniques and brushes. As you can see in this painting there's short smooth fur, long wispy fur, thick clumpy fur, etc.
In most cases when painting fur you want to have at least 3 layers, the darkest colors being the deepest areas closest to the skin, ending with the lightest colors highlighting the tips of the fur. You'll also generally want to start with the thicker brushes on the bottom layers and finish with thinner, finer brushes for the wispy hair tips.
In this first detail (below) we have broad areas of relatively smooth fur. Use a fan brush to build up layers. Start with a larger fan to build up the bottom layer and switch to a smaller fan on the next layer. A filbert rake brush can also work here, depending on the animal and its fur type. Try to vary your brush stroke length, stroke angle, and brush angle to keep it from looking too uniform. Wild animals don't get their fur brushed so you want it to look a little scruffy. (Of course, if you're doing a pet portrait of a prize show dog then this doesn't apply. You'll want your brush strokes to stay pretty uniform so your pooch looks well-groomed.) Once the first two layers are down go back and add a few strokes here and there with a script liner to add a little more scruffiness and highlights.
In this second detail (below) we have areas of clumpy fur. Here start with a medium round brush to block in the basic shape of the clumps. Again, start with darker colors on the lower layer. Then use a smaller round to do the middle, lighter layer. Finally, go back and add some wispy fur tips with the script liner once again to the top, lightest layer. Always try to vary your stroke lengths and angles, even getting a few wiggly lines in there once in a while, to give it that "wild" ungroomed look.
Nothing really magic about the technique. It just takes remembering to go from dark to light, larger brush to thinner brush, keeping your strokes varied. And a lot of time and patience. :)