For years, whenever I made any kind of stew or meat dish that required first braising the meat and then thickening the sauce, I used the flour and water method—mix some flour with some water, whisk or shake well to remove lumps, and slowly pour into the hot broth a few minutes before the dish is done. Now, I used this because my mother used this method, and she inherited this practice from her mother. It is an effective method, not entirely without risk in the lump category, but I always had to make sure I had cooked the broth down enough so that the dilution that resulted from adding this water/flour mixture would not seriously weaken the flavor of the final product. This was always a balancing act, something of a guessing game, and did not always turn out to my liking.
Enter Beurre Manie—this is a mixture of equal parts of butter and flour, creamed together to make a paste. You then would toss in small bits of this mixture into your bubbling stew, and the butter would melt, releasing the flour into the sauce, beautifully thickening in the process, and without diluting the broth. As with the water and flour method, you need to reserve a few minutes of cooking time to be sure you get rid of that raw flour taste. And if you have made too much of the paste, just store in your refrigerator for the next use.
Roux is another way of using this equal parts butter and flour, but is used to prepare a sauce from the base up, melting the butter before adding the flour, cooking for a few minutes before adding your other hot liquids. The following link provides a good explanation of the uses of each method: