Articulation materials are those materials placed on woodwind instruments at any point where two pieces of key work engage with one another. Any time one key or lever activates another key or lever, a piece of articulation material will always 1) prevent noisy metal-to-metal contact and 2) sometimes allow adjustment of the relationship between the two keys. For instance, if pressing one key closes another key, they will both need to close at exactly the same time. Placing material of the correct thickness at their articulation point will allow that. If the material is too thick or thin, one key will close first and prevent the second key from closing. In cases where adjustment screws exist, the articulation material just silences the connection, and changing the position of the screw is what changes the relationship between the keys.
Articulation points are different from key feet, which are the parts of the keys that contact the body. Key foot materials tend to be chosen exclusively for their firmness or noise level, while there are other factors that come in to play when selecting articulation materials including compression and coefficient of friction. There are a lot of materials useful for articulation points, some better suited to certain purposes than others. These are some of the ones I use.
Natural cork: is very quiet but squishy, so thicker pieces don't hold adjustments well. If you have a strong grip on the keys, you'll likely squeeze through any compression so the drawback of "squishiness" is, in that case, negligible. Natural cork has a relatively high coefficient of friction and tends to tear easily, so it's not great for sliding articulation points or very small points. However because it's so quiet, it's often found under the articulation screws on oboes. In this application the corks should be replaced at least once a year lest they become too compressed and start to get noisy, or tear. When it is used in sliding points, natural cork can make a squeaking sound as it moves. Nonetheless, it's the material most often used by manufacturers on their factory set-ups because it's inexpensive and forgiving of sloppy pad work. It's the material I use least often.
Gummi cork: is firm and resistant to tearing, but its stability comes at the cost of being noisy. Gummi corks are not to be confused with low-grade composition cork. Gummi cork is made of very small scraps of cork held together with very strong glue and cut to a precise thickness. Because of its stability it's great for places like bridge keys on clarinets and oboes. It's also commonly found on the backbar of saxophone F# keys, but here the noisiness is more noticeable because saxophone keys have more mass that's slamming down on to the cork. It is extremely stable, and so very useful for joints where a precise regulation is needed. Gummi cork isn't exactly a low-friction material, although it does last longer than natural cork when used in sliding joints. Over time, though, it will compress or tear in certain application, like the back of the F/C key on clarinets, where the key contacts the left hand F/C lever.
Felt: is very quiet and often resistant to tearing, but is also very compressible and not great for precise adjustments. Felt can't be made especially thin either, so the right thickness isn't always available. Felt can be soaked in a variety of substances to harden it and make it more dimensionally stable, sometimes without making it too much noisier. It has very low friction and can work quite well for sliding joints, especially where precise adjustment isn't required. It's compressibility makes it good at absorbing shocks, so it's useful under saxophone key feet when one is trying to eliminate "key bounce," though over time it can permanently compress and become too dense and firm to prevent bounce.
Synthetic felt: is more stable and firmer than felt, but it's also a little bit noisier, and a good bit more expensive. It great for joints where a precise and quiet connection is needed. Again, though, it's not available in especially thin sizes so it has somewhat limited applications. I often use it in places on higher end instruments where tradition dictates that felt be used - between the A key pearl and Bb bis key on saxophones, or between the thumb Bb lever and B key on flutes. Because it doesn't compress as readily as felt, it's a longer-lasting fix for eliminating key bounce.
Nylon: most often used as an insert in regulation screws on flutes, oboes, and clarinets, nylon is relatively quiet on small articulation points (such as you'd find on those instruments). It's not as quiet as natural cork or felt, but on small keys with little range of motion, the noise can be quite manageable. The greatest benefit is that nylon tips are extremely stable and long lasting, perhaps never needing replacement as long as they're not damaged. They can't fall off like pieces of cork or felt and will remain dimensionally stable regardless of how often the screw is moved. Nylon is also fairly low in friction, so it doesn't stick or cause mechanisms to "hang up."
Leather: very thin pieces of leather are sometimes found under the articulation screws on oboes. Specifically, the leather is usually of the type used in making saxophone pads, which can be a little too thick for this application on certain instruments. It's resistant to tearing and relatively quiet. It has a little more friction than other materials, but since the articulation points on oboes don't really slide, that is of minor concern.
Teflon sheet: is of course very low in friction, so it's an excellent choice for any sort of sliding joint. Teflon sheet is available with a treated back that can be bonded to keys with glue (the front surface, of course, won't stick to most adhesives). It has a plastic-like consistency, so it's also very stable and resistant to tearing. Unfortunately it's also very hard and loud, so its use is always a bit of a compromise. Again, on smaller contact points, the noise is less noticeable because of the small mass of the keys.
Ultrasuede: is low in friction and extremely resistant to tearing. It's not very firm, though, so it's most often my choice for sliding joints where a precise adjustment isn't needed. For instance, on old baritone saxophones where the Low B and Bb pads are separate from their levers, linked by a sliding ramp. Because ultrasuede isn't very dense it squishes easily, but doesn't stay compressed. Ultrasuede is pretty expensive, like synthetic felt, and a royal pain to cut with a razor blade.
Laminated cork: is made from a sheet of very thin paper laminated between two pieces of very thin cork. The idea is to create a thin cork (.003") that will be resistant to tearing or compression. Generally, it fulfills those criteria, but any material that thin is going to be somewhat noisy.
Sorbothane: is often used in the insoles of shoes, but has found limited use among musical instruments. It is excellent at absorbing shocks, so it's great for eliminating "key bounce." It's also pretty quiet, but it's designed to compress and recover, so while it always returns to its original thickness like ultrasuede, it can't be used for precise adjustment or anywhere that a firm feel is desired. It's also a pain to cut.
Other: Whatever is out there! A lot of technicians get creative with the articulation materials they use, and find things that work better for their particular needs and the needs of their customers. I've lately been trying out pieces of very thin natural cork with a piece of Tyvek envelope laminated on top of it, for any place where a very thin piece of material is needed, and especially under regulation screws. The Tyvek is almost impossible to tear, it's slippery, and it's extremely thin. Adding a backing of natural cork cuts down on the noise of the Tvyek (being very thin, it's also very noisy), and since the Tyvek itself is what makes contact, it protects the cork underneath from tearing.
Really, the best articulation material is what makes the customer feel comfortable with their instruments, so a technician must be capable of using all of them properly. As always, it is critical that a technician understand the tools at their disposal and be prepared to deploy them in order to suit the customer's specific needs and preferences.