There is always an element of doubt for every guitarist if he/she needs an active or a pasive pick up. Frankly speaking i think you can come to conclusion pretty easily if you are clear about the kind music that you would like to play. Basically active pick ups are generally used for metal guitarists who are in need of high gain to get those extremely heavy and distorted sounds or if you need the extremely low sounds like maybe a guitar tuned to c# drop.
If you are unable to come to a conclusion on what kind of music you would like then its better you end up acquiring a passive pick up. Passive pick ups basically give you a more crunchy sound.
Now that you have the basic concept of these two we shall see as to how it actually works:
Active pick ups: They are basically those pick ups which have low inductance. They act as pre amps and work on a 9v battery. Sometimes even two.
Passive pick ups: With these we can get more crunchy sounds that will be influenced by the tone and volume controls on you guitar.This is basically a magnet with lots of turnings on it and there is no pre amp so there is no necessity for a battery. As a consequence of this we are unable to reach really high or low frequencies.
Most metal guitarist tend to use active pick ups to get those heavy sounds. Its like a boost to your amplifier and your processor tone. For active pick ups one should look at no other company other than EMG. In case you are looking for passive pick ups it is better that you get your hands on maybe a dimarzio or a seymour duncan. If your guitar does not have a really impressive body then some people do suggest seymour duncan active pick ups in case you are in need of that high gain. However, even if you do get the active pick up it is more of a waste as it is equally important for your guitar body to have a wood that suits and is meant for heavy sounds.
For example a basswood body guitar with EMG 81 pick ups and a mahogany body guitar with the same set of pick ups which one do you think will sound better. Well without a doubt the guitar with the mahogany body will sound better. In case you need some great heavy sounds I suggest you first buy a guitar with either mahogany or alder wood with maybe a maple neck and a ebony or rosewood fretboard. After that for a finer and a heavier sound or even a crunchier sound you can add on the pick up of your choice.