The total atmospheric optical depth consists of three basic parts: molecular (Rayleigh) scattering by the atmosphere, atmospheric absorption by gases such as ozone, and scattering by aerosols (aerosol optical thickness, or AOT). The figure in the attachment shows the relationship between percent transmission and optical thickness.
Very high AOD = very high aerosol load (as Muhammad said). What you find is typical of the Gangetic basin during pollution or dist episodes. There is no absolute limit. Around deserts, values above 1 (and usually below 3) are typical of sand storms, for instance.
More info on AOD=1. Since we started measuring it, we never got AOD=1 in a Mediterranean coastal site. In 2012 we had a strong wildfire some 50 km away and a Saharan dust intrusion at the same time. You could see the smoke looking through the window but the maximum AOD we got was 0.78
AOD value of 1 indicates hazy conditions and AOD value above 2 or 3 represents very high concentrations of aerosols in the atmosphere during severe pollution.
Some answers above insist on high AOD caused by "pollution", but the fact of the matter is that the largest AODs (above 5) are caused by smoke or dust storms.