Drums are designed to resonate. Damping supresses resonance. Something wrong with this picture? Yup. When you damp a drum you are killing much of the tone it produces, as well as reducing the volume and projection.. You will often find that a bit of a ring helps your sound to carry too. So damping is the last thing you should do if your drums ring too much

Having said that, there are the odd occasions when damping is necessary to kill a small amount of ring. Here follow some common techniques to achieve the required sound:

Tuning
Careful tuning can preserve the pitch while decreasing sustain, ring, and overtones. This usually involves carefully detuning the head at some points, while raising the pitch at other points. Depending on the amount of muffling you want, you can detune these points or use them to retain the pitch by tuning them higher.

Heads
Often overlooked, but different kinds of heads exhibit different damping properties. A thicker twin-ply head will not ring as much as a single-ply. And hydraulic heads - well, are hydraulic heads, and sound as if you are playing underwater. Some snare batter and bass drum heads heads have e-rings built in.

O-rings, zero rings, or donuts
These are small rings made from the same material as the heads. They rest on top of the head with their outside diameter large enough so it just fits inside the rim. The inside diameter can be as large as you want, depending on the amount of muffling needed. The nice thing about these items is that they reduce the sustain of the drum without killing it totally. They also preserve the volume of the drum. Due to their even nature (same width all the way around), pitch and tone also stay constant in different places around the head.

Piece of cloth
Real emergency measure here. Lay a small piece of cloth or tissue on the head, next to the rim, and tape it down. Unfortunately, you will find the tone of the drum changes in different places around the head. OK if you like that 50's kind of thud tom sound.

Built-in Damping Mechanisms
Some older drum kits came with an internal or external mechanism which push a felt pad against the head. Remove these - they cause all kinds of problems: uneven tone and pitch, rattles and buzzes etc. They are also just another piece of metal to weigh the shells down with extra weight.