Both of these things are expensive to buy in stores and sometimes completely unavailable commercially. I haven't come across duck stock for sale anywhere, although chicken stock is ubiquitous. Duck stock tastes a little like chicken stock but it is richer and darker. It's wonderful to use for risottos, soups and sauces. Duck flavours go particularly well with mushrooms, potato, beetroot and any type of poultry where it can impart a complexity of flavour beyond that of the original bird. For this reason, it's great to have a carton of duck stock in your freezer - it keeps very well frozen and you'll always have it available.
Duck fat is even MORE useful. It has a high smoke point which means it doesn't burn like butter. You can use it for both shallow and deep frying. If you've used a large quantity you can strain it afterwards and re-use it. Roasting vegetables, particularly potatoes, in duck fat is well known to contribute both to their flavour and their crispiness. It can be used in salad dressings (you need to heat it slightly to make it liquid) and in making pastry, where it gives a fluffy light crispness. If you'd like some more detailed information on its uses, check here.
I hope I've convinced you of the usefulness of duck fat and stock - the best bit is that they take very little work to make if you have cooked a duck. (Obviously, they're a bit trickier if you haven't.) Basically you have to make both concurrently. Here's how.
How to clarify duck stock and render duck fat
After you have eaten your delicious duck and cleared the table, scrape all the bones, meat scraps, skin, etc into a saucepan. If you have cut off any parts of the duck prior to cooking (the neck or some of the fatty skin for example), reserve these at the time and add them in to the saucepan at this point. Add enough water to cover amply all of the bones, and put the saucepan on to simmer while you do the normal after dinner things (having dessert, washing the dishes, stacking the dishwasher, etc.)
At some point (I am assuming you have cooked and eaten your duck in the evening, for dinner) you will want to go to bed. Hopefully this will be at least an hour, possibly rather more, since you put the duck leftovers to simmer in the water. Before you go to bed, pass the duck mixture through a fine sieve into a ceramic or metal bowl. The sieve should catch all of the bones and solids while allowing the liquids to pass into the bowl. Discard all of the bones and solids. Cover the bowl with cling film and put it in the fridge overnight.
In the morning (or the following night, if you have to rush off to work first thing), you should find that your bowl of duck stock has magically separated, pushing the cream-coloured fat to the surface and creating a firm, clear jelly underneath.
If the fat is still liquid, put it back in the fridge for a while. You need it to be firm. It won't go hard exactly, but it will become pale and solid enough to scoop up in a spoon.
Heat up both your bowls - the stock and the fat - to encourage the magical 'separation' process for both. This will help you get the last bits of fat out of the stock, and the last bits of stock out of the fat. Leave them both to cool - this may take a few more hours; if you do the first stage in the morning, leave the bowls in the fridge until the evening and then do the next stage.
When the remaining fat has again returned to the surface of the stock and has gone pale and somewhat solid, use a spatula or spoon to scrape it all away. You should end up with a clear jellied stock, with no fatty film on the surface.
Heat until liquid, then transfer this to a bottle or lidded container and freeze it. You have made a beautiful, clarified duck stock!
You should end up with a container full of very pure, creamy white duck fat. Seal the container and put it in the fridge. You now have wonderful, rendered duck fat!
That's all - I hope you enjoy this relatively simple process and get absolutely everything out of your duck!