It's actually spelled 'roux', and is a mixture of oil and flour, cooked to remove the starchy taste of the flour.
It's a great thickener any time that you don't need the sauce to be clear, and you have time to cook it down.
I typically use it for cream sauce (including cheese sauces, such as for mac & cheese) and gravies.
As for benefits -- it's habit at this point, so I'm not really sure --
I guess it's made from things I always have on hand.
You can also get flavor from the roux, if you cook it longer , but you'll adversely affect the thickening ability.
Gumbo is normally made from a dark roux (the Cajuns have a series of names for the color of roux, including 'brick', 'peanut-butter' and 'chocolate')
cooking.stackexchange.com/users/67/Joe
A "Roux" is a mixture of 50% butter, 50% flour that is used as a starch thickener for a number of "mother" sauces (notably Béchamel, Espagnole, Velouté).
For a white sauce base, you may heat both butter and flour together in a saucepan over a low flame while combining with a wooden spatula.
After just 30 seconds mixing, you will get a consistent semi-liquid that is your "Roux".
Now, turn the heat to medium and continue stirring rapidly.
Gradually add milk, cream or other liquid as required.
As you add more liquid you can slow down your stirring speed.
The sauce should thicken in just a few minutes.
For a dark sauce base, heat the butter first on a medium/low heat for about 5 minutes until the butter takes on a nutty flavor.
It will also darken due to the sugars caramelizing.
Once you have the desired color, add the flour and continue as above.
See Sauces for more information.
cooking.stackexchange.com/users/335/octonion
as mentioned by others a 'roux' is flour and oil/butter mixed together, while under heat.
a roux is normally used for thickening sauces (usually cream/cheese type sauces).
the benefits of using a roux, are that your sauces will not get lumpy.
Try just adding flour directly to the sauce next time.
All you will get are lumps of flour in it.
Not nice!
the roux keep everything nice and smooth.
Same thing can be said about thickening with corn starch.
You mix corn starch into water (or broth) first.
Why?
because if you directly add the cornstarch to the sauce, all you get is cornstarch lumps
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Typical procedure: Heat shortening (oil or butter), add flour, cook, stirring continually, for a few minutes.
Add liquid a little bit at a time.
Many sauce recipes (e.g. Mac & Cheese white sauce) are just built up this way; in other cases you'll in turn add the somewhat thinned roux to something else.
cooking.stackexchange.com/users/149/bmargulies
You can get roux in a jar You can also make roux in the microwave
You can even make a dry roux without any oil!
The most difficult part of making a roux on top of the stove is that you could burn it and have to start all over again.
cooking.stackexchange.com/users/25/Cade Roux
For making gravy I prefer to make dark roux in the oven.
Melt 1 part butter in a baking dish, sprinkle with 1 part flour, bake for 30 minutes (stirring and re-spreading half way through).
This works great once you have taken the turkey or roast out of the oven to rest.
Once you have your roux to the desired color starting adding your drippings and broth for a great gravy.
cooking.stackexchange.com/users/51/Janelle
Many have already answered: What exactly is a roux and when is it used.
I would only like to add more to answer your question of "what are the benefits of using it versus other cooking methods?"
As many have also said smoothness and uniformity of texture.
I would also add "Mouth feel" and flavor.
Most other thickening agents will not have the same "Mouth Feel".
Especially corn starch which tends to be more like a "Gel".
If thickening agents were paper, roux would be "matte" and corn starch would be "extra glossy".
lol
Also the final flavor and color of roux can be dramatically altered merely by adjusting the cook time.
It is truly a fundamental flavor skill to master.
I encourage you to buy some starches and thickening agents and make some small batches and taste for yourself the differences.
It is a great way to sharpen your palette.
cooking.stackexchange.com/users/None/
I've had excellent Macaroni and Cheese that I was told was made with a "roux."  What exactly is a roux, when is it used and what are the benefits of using it versus other cooking methods?
What is a "roux" and what are its uses?
cooking.stackexchange.com
<culinary-uses><roux>
cooking.stackexchange.com/questions/1089
cooking.stackexchange.com/users/128/Oren Hizkiya
A roux is a 1:1 mixture of flour and oil or butter, cooked to remove the starchy taste of the flour. It is used as a base for a number of "mother" sauces or to thicken sauces that don't need to be clear. A roux is a mixture of butter and flour and is used to thicken sauces.
A roux is a mixture of butter and flour mixed together under heat. A roux is used for thickening sauces.