This answer is courtesy of Ellen Zweibel, University of Colorado.
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When we say that a wave dissipates, we mean that energy carried in organized form by the wave is transformed, irreversibly, into heat in the substance which carries the wave. For example, the energy carried by a sound wave is detectable as a pressure force that causes the eardrum to vibrate. In ultrasound therapy for a sports injury, the sore area is exposed to sound waves which are too high in frequency to be heard, but which are absorbed by the body and heat the injured tissue.
Alfven waves and slow mode waves both require an ionized gas, with a
magnetic field, in order to propagate, but they are otherwise quite
different. Slow mode waves (and fast mode waves) compress the gas (as
ordinary sound waves do) and also bend the magnetic fieldlines. Alfven
waves just bend the fieldlines without compressing the gas. It turns
out that magnetic fieldlines behave very much like elastic
strings. Alfven waves are like the waves excited by plucking a string.
In many astrophysical systems, including the solar atmosphere, when
you compress gas it increases the rate at which it radiates light. In
a sound wave, a fast mode wave, or a slow mode wave, some of the wave
energy which goes into compressing the gas is lost in the form of
light (or photons). Energy is also lost by thermal conduction (a hot
and a cold substance in contact equalize their temperatures because of
conduction). Because Alfven waves do not compress the gas, they do not
lose energy in this manner. Therefore, they dissipate more slowly."
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