First, a DI is an impedance matching device. It's designed to allow a high impedance input (aka, an electric guitar, most synth outputs, etc.) plug into a low impedance input.
They take into account the differences between line and low (mic or guitar p'up) level using a pad which attenuates the input, generally between 20db-40db.
To your question of the difference between passive and active, power is the simple, but incomplete answer.
Passive DI's convert high impedance to low impedance using a transformer. No additional power (beyond the input signal) necessary to operate.
Active DI's use electronic circuits to convert high to low impedance and require batteries or phantom power, provided by the PA mixing console or a separate, phantom power supply box. Some active circuits require more power because they are used as signal processing and have their own power supplies that must be plugged into a wall socket.
But most importantly, passive DI's will lose some treble in the conversion. Few passive DI's are available that compensate for this. Active DI's will have a more transparent frequency response (though each certainly has its' own character) and, therefore, is preferred for more articulation. They have more likelyhood of noise issues, however, due to the additional electronics and use of phantom power from the console. Good passive DI's are also nearly indestructable. Most active DI's, while built like tanks, are not quite so robust as their passive siblings.
Radial makes a line of active and passive DI's that are extremely high quality, sound great, and while expensive, are not boutique items, per se.
If you're on a budget, A/B some DI's into a PA at a store. If you hear a difference, by the one that sounds better. If you don't, buy a relatively inexpensive passive DI. It should last you a lifetime unless you desire more open, articulate sound available from high quality DI's.