Various types of stars have dust around them. In the well known case of the Pleiades the dust around the stars is unrelated to the stars, and is instead dust from an interstellar cloud that the Pleiades cluster has drifted into on its way through the galactic disk. This is seen in the picture below courtesy of Robert Gendler (see his home page for nice pictures).
The dust in interstellar space is observed in a variety of ways: due to its passive absorption of starlight ("extinction"), by its production of polarization in starlight, and then at long wavelengths due to its thermal radiation. Most of the dust in the interstellar medium is very cold and only radiates at at wavelengths beyond 100 microns. Where there are hot stars in and around molecular clouds the dust gets heated to much higher temperatures (say 75 to 150 K) than in the general diffuse interstellar medium, and its radiation can be detected down to around a wavelength of 10 microns or so. I do not study the dust in the interstellar medium, but it is an area of active research with many interesting results.
I am more interested in stars with associated circumstellar dust that the star itself has ejected as part of a stellar wind. Such stars are generally cool stars: M-type giants and supergiants for example, or carbon stars. We usually observe the dust around these stars via direct thermal emission of radiation from the dust. Dust temperatures around the evolved stars range up to somewhere in the range 1000 to 1500 K, and thus the dust radiates strongly at all wavelengths from 3 microns to beyond 100 microns.
The image below shows sky survey images for a specific region of the sky at wavelengths from the optical (in the blue and then in the red parts of the spectrum) as well as at longer wavelengths J, H, and K courtesy of the 2MASS sky survey. A normal star at upper right is marked on the various images with the blue arrows and the magnitudes for that star are listed. The star of interest here, CW Leo = IRC+10216, is at the center of the frames. It is entirely invisible in the blue part of the optical wavelength range, its seen as a faint star in the red part of the optical, and its gets much brighter through to K-band. This continues on to the 5 to 10 micron range where its the brightest star in the sky.
For those who are astronomers and can think in terms of magnitudes the following comparison shows how big an effect the dust absorption has on the emitted spectrum in this object. In the IRAS  filter this object has a magnitude of about -7.5, so whereas alpha CMa has B magnitude of -1.46 and IRAS  filter magnitude of -1.35, so the B -  colour value is -0.09. IRC+10216 has B > 20 and IRAS  of -7.5 so B -  is > 27 magnitudes. The difference is entirely due to the star having a thick dust shell around it which absorbs the star light and re-radiates it at long wavelengths.
The other star in the field that I am using for comparison is probably a normal K-type giant star, similar to Aldebaran for example, judging from the change in magnitude from B to K bands. Thus its a fairly cool star, but CW Leo has a much, much cooler spectrum. It looks like roughly a 500 K blackbody source; yet that is far too low a temperature to be reasonable for a star. Thus we deduce that we are seeing not the starlight but rather the radiation from a thick shell of cool dust around the star itself. In this case we know from other observations that the star inside the dust shell is a carbon star, one that by chance is relatively close to the Sun.
Looking at the spectral energy distribution of this object, in the Figure just below, I compare the observed radiation from its Infrared Space Observatory spectrum and the J, H, and K band photometry for this star to what I estimate the bare star would look like without any dust around it. The "bare star" spectrum is a carbon-star template spectrum formed from optical, near-infrared, and ISO spectroscopy of some carbon stars, scaled to give the same total observed energy at the Earth as the IRC+10216 spectrum. The estimated V magnitude is about +2.4, so it would be a fairly bright star but not one of the brightest in the sky. However it would be a long period Mira variable, and if the amplitude were high enough it could vary between being one of the brightest stars in the sky at maximum and being invisible at minimum, which would be quite interesting to see.
In these dusty evolved stars we observe a variety of dust features in the 3 to 50 micron wavelength range. Some of these features are securely identified. Other features have identifications that are less secure, or in a few cases have no identification. The pages linked below show first the infrared spectra of some stars without dust, which for the cooler stars show molecular absorption bands, and then example spectra of stars with dust.
"bare" stellar spectra in the mid-infrared
the most common features: silicate and SiC dust
less common features: extreme carbon stars, the AlO/silicate complex, unusual silicate features
dusty HII regions, planetary nebulae, and related objects
unusual features: crystalline silicates, ice bands, and featureless cool dust emission