Passives require an amplifier, whether it be in the form of a powered mixer or a separate amplifier. Personally, I'd rather use a separate amp since powered mixers can be quite expensive for a particular output wattage. Worse, if the powered mixer breaks down, you lose 2 components in your system... it's usually much easier to replace a separate power amp.
Active monitors have amplifiers built in so you don't have to worry about matching amps to speakers. Some prefer these because you don't have to route bulky speaker wires to passive monitors. The downside is that actives tend to be more expensive than passives.
Full range means that a speaker can replicate all audio frequencies. Because of physics, bass frequencies are usually handled well by woofers and high frequencies by tweeters. A full range cabinet usually contains 2, 3, or more combinations of woofers, mid-range and tweeter drivers.
