The knee is one of the most complex joint in the human body.
The patella or colloquially known as the knee cap is one of the tendons in the knee that is responsible for knee extension.
Badminton places a lot of movement loading on the patella especially movement like jumping and lunging.
For example a Knee supporter from Yonex. This model covers the entire joint and is made of Neoprene, which allows stretching and is generally warm on skin contact.The cons of a full support is it feels restrictive a bit when moving around.
A more popular type of knee support is the simple patella brace, for example this model by LP SUPPORT. This model is made up of neoprene and can be tied more snuggly with a velcro fastener. This is good especially for players who like to jump a lot. This can be worn as a precaution during game to prevent jumper's knee...
Victor also made a very good product, a knee belt (as Victor calls it). This particular model as a tube like structure embedded within the neoprene which fits very comfortably on the knee cap. The velcro is used to fasten it. This model is as good as the above, except that it may not be tied as snuggly.
I prefer the patella brace compared to the full knee support, owing to better comfort and mobility. The support of the patella is generally sufficient for badminton game. For more serious injury, consultation with a sports/orthopaedic doctor is paramount.