Air conditioner units are capable of living long lives, but when the unit is stored for extended periods during the winter, who-knows-what can get inside of it. Both a unit stored in humid conditions and an older unit can blow all kinds of impurities through the front vents upon summer startup. The causes vary but can be remedied.
If an air conditioner isn't clean and sat in a garage somewhere for months or years, it's likely that a lot of dirt and dust has settled within the unit. If black dust blows out upon activation, turn the unit off, open up the panels leading to the different components of the air conditioner and clean the interior with compressed air or a mild cleaner and cloth. The fan could be part of the problem if a hefty amount of dirty rests on the blades.
Deteriorating duct liners produce black particles, which blow from the vents and collect on walls or the ground. A technician can tell you if the problem is the duct liner. If the duct liner is the source of the black dust, it should be replaced right away, as the particles could be dangerous if inhaled. A really old unit might house an old or disintegrated air filter. When the fan blows after the unit is turned on for the first time in years, parts of the rotted air filter blow out of the cooling vents. Not only will this dirty the room, it allows impurities to flow freely through the AC's components, opening the unit up to clogged components and eventual repairs. The front grille of the AC unit opens by hand to reveal the filter. If the filter is obviously old or filthy, replace it with a new one or clean it.
It's possible that a part of the air conditioner, such as a the heat exchanger, could have burned up and produced soot. If the dust seems like small ash particles, turn the unit off immediately and call a technician to look at the unit.
An air conditioner sitting in humid conditions sets up a prime breeding ground for mold. The fan inside the unit might blow harmful bits of mold into the room upon startup. If the unit had a malfunctioning fan the last time it was used, then mold can easily grow around the fan components. The front panels of the unit can be removed to reveal the fan motor and fan. This is a good time to check the interior for debris that might also be the cause of the black dust. Remove any debris and clean the mold from around the fan motor parts. If necessary, you can take off the fan to better clean it and then reinstall it.