I'm basing my answer from this WikiHow article as well as those two french article consoglobr and sanscroquettesfixes.
Here is what you should remember about them:
By the aspect
Real fur has a leather base. Generally, the real fur is thick, with a part of fine hairs more curly, and longs hairs that divide when you blow on it.
Most faux fur products have the same length and color of hair, although some imitate the real fur. The real fur, however, may have been cut to the same length for a uniform look, so you need other tests.
By the touch
For faux fur, the hairs are often rough.
Real fur is soft and supple and it's easy to wrap the hair around your fingers.
Fur: feels very soft to the touch, falls in a smooth and sleek line, passes through your fingers as if you're petting a cat.
Faux fur: feels coarse and rough to touch, feels synthetic; can be sticky to touch in humid weather, and it might have the same feel as a stuffed toy animal. It might stick to your hands if it's made out of a plastic-y material (this only works if your hands are sweaty).
With the burn test
You can pull a few hairs from the item and put them on top of a flame.
The real fur will burn in the same way as your hair and the smell will be relatively the same.
For faux fur, we will observe a melting and smell similar to that of plastic.
With the needle test
Stick a pin into the item (through the fur and its lining).
If it is hard to push through, or resists completely, it is likely to be real fur, as you're trying to push through the leather lining to which the fur remains attached.
If it goes through easily, this suggests it is faux fur because the pin is sliding through a synthetic base.