Put simply, active speakers are full range speakers, or subwoofers that have an amplifier module built-in to them. This means you no longer need to carry an external power amplifier around with you. This can greatly reduce the amount of time it takes to set-up and tear-down at gigs.
Active speakers are also sometimes called powered speakers, or self-powered speakers. Whatever you decide to call them, it is all the same at the end.
One thing to love about active speakers is that every manufacturer of active speakers knows that some gigs are going to require subwoofers. So, many speaker companies make active subwoofers that perfectly match their active full range speakers. As far as setting up a pro audio sound system goes, it doesn't get any easier than that. I do suggest that you use a dedicated external electronic crossover, but you can run signal from a subwoofer stright to a full range active speaker if you need to. Many active subwoofers have a dedicated output just for this purpose.
One final note. When active speakers came out over a decade ago, I was concerned that they would not hold up. I thought that the acoustic energy of the speaker components would destroy the amplifier modules, and for a time I was right. That problem is long gone, and active speakers are now every bit as reliable as their passive speaker / external power amplifier partners.