Confusion is easy in this area as there simply is no proper terminology in consistent use.
To try and make this clear, 'powered' and 'active' are terms that define how a loudspeaker is driven. As pointed out above, 'powered' speakers are simply normal passive speakers that have a single amplifier built in, ie passive speaker plus single channel amplifier in one box. In hi-fi use it is common to put a 2 channel amplifier in one speaker, the second channel drives the second speaker via a length of regular speaker cables in what is known as a 'master/slave' configurations.
Active speakers on the other hand have no passive crossovers, the frequencies are divided by an active crossover that comes before the power amplifiers, one for each frequency band. These are single, individual and identical pieces.
The real confusion starts when you want to add functionality, say pre-amp functions, Bluetooth or what have you. For sound technical reasons, it is easier to implement this when the speakers are in a 'master/slave' configuration, so most 'full function' speakers tend to be powered, rather than active. The obvious exception being the AVI ADM9/10 series which are possibly unique in being fully active and configured as a 'master/slave'.
Most, though not all actives, tend to be simply line in, ie you will need a pre-amp of some sort to drive them, check out the description of the new AE1 Active in another thread.
So, to sum up, powered systems with added functionality like remote control and Bluetooth are rather like a sort of 'grown up' docking system, actives tend to be stand alone speakers, designed in the main to be run of a pre-amp, computer or in the pro-world, some sort of mixing desk.