You can't run any speakers without an amp.
Powered Speakers have a built in amp that drives the drivers via a crossover.
Active Speakers have at least two built in amps. The signal from your mixer / whatever goes straight into an active crossover which splits the signal into highs and lows, and sometimes mids. Those signals are then sent to individual amps that only feed their own section.
Passive Speakers are just the drivers, and usually a passive crossover, in one cabinet. A sepereate amp (or two) is required.
Powered Speakers and Active Speakers are much heavier than passives but you save space as there is no need for a seperate amp. Also no speaker cabling is required which simplifies the setup somewhat. Active speakers have the crossover, amps and drivers specifically matched to each other to achieve the best performance.
Passive speakers are more complicated to set up as you need the right amp to suits the speakers. If you want to use a sub that complicates things further. But if you blow a speaker or amp, you only have to replace that part, not the whole thing like you would with a busted active speaker.
In summary, Active speakers are simple and are already set up for good performance, hence why many people use them. Passives with a seperate amp are more difficult to set up correctly, but in the right hands are probably better than actives of an equal price. Passive systems are easier to upgrade or expand too. But as you're having to ask what's best, that's probably not you (no offense).
What do you want these speakers for?
If it's just for home use than active is probably the way forward. KRK or M-Audio are popular on the budget end of the scale. If you want something for gigs, there isn't as much active stuff on offer. The Matrix system is a small active system that's popular. Otherwise PA systems tend to use passive cabs.
I hope this helps!