There are two basic actions which constitute the prohibition of Kibus (laundering) on Shabbat. First, the very act of applying water to a garment constitutes laundering-"Shriyato Zehu Kibuso." The second act is wringing out water from a garment-"Sehita".
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) in Siman 302:39 outlines three opinions in the Rishonim (early authorities) as to which garments are subject to the prohibition of Kibus (laundering). The first opinion holds that Kibus only applies to a soiled garment. There is no Torah prohibition to launder a clean garment. The second position maintains that even clean garments are subject to Kibus. However, there is no problem if the water is applied "Derech Lichluch"-in a manner that soils the garment, instead of cleaning it. For example, it is permitted to dry one’s hands on a towel, since the water from the hands does not clean the towel, but rather dirties it. The third opinion is more stringent and prohibits applying large quantities of water, even to a clean garment, even "Derech Lichluch"-in a manner that soils it.
Hacham Ovadia and Hacham Bension both hold that Maran’s opinion is in accordance with the first, more lenient, opinion that Kibus only applies to a soiled garment. Therefore, Hacham Bension permits preparing a cold compress for someone with fever by dousing a clean towel with water, so long as the rag does not become soaked and present a potential problem that he will wring it out. Hacham Ovadia presents an additional application of this leniency. If one end of a garment is set on fire, he permits dousing the other end with water so that the blaze will be extinguished.
Maran (302:19) writes that it is "good" to shake the excess water off the hands before drying them. Seemingly, this is in accordance with the third opinion that large quantities of water constitute laundering, even when drying hands on a clean garment. However, the Magen Abraham (Rav Abraham Gombiner, Poland, 1637-1682) explains that it is not the letter of the law, but a recommended Humra that takes the third opinion into account when washing hands, as it is possible that the excess water constitutes a "large amount."
It is important to note, that sometimes at large meals, many people dry their hands on the same towel. By the time the last people use the towel it is already soaking wet, and "drying" their hands would be a problem of "Sehita"-wringing out the water from the towel. Therefore, it is advisable in such cases to provide several towels or use disposable towels.
The Halacha permits dousing a clean towel with water. Therefore, it is permissible to prepare a cold compress on Shabbat. However, it is preferable that one shake off the excess water from his hands before drying them. One should be careful not to dry his hands on a towel that it already soaking wet.