Active and passive DIs both convert high-impedance unbalanced or balanced signals from passive instruments or other devices (active instruments, mixers, etc.) to low-impedance balanced signals. In the case of passive instruments, the signal can travel down long cable runs with no loss of signal quality. In the case of other signals, the balancing of the signal keeps noise levels down in long cable runs. DIs can also isolate two pieces from each other electronically, reducing noise from ground loops, like hooking a DJ mixer or bass amp onstage to a mixer at FOH.
Active Dis use a powered circuit to do this, either battery or phantom powered. Some even use AC power.
Passives use a transformer.
Active DIs have very high input impedance, making them great for instruments with passive pickups.
Passive DIs have a lower input impedance, and loss of tone might happen if you plug, say, a strat into a passive DI.
Passive DIs handle transients better when pushed hard, and can handle hotter signals for the most part, than actives. The transformer doesn't clip like active circuits, or have strict limits on input level like an active. But if saturated by too much signal, loss of quality can result, usually the loss of low-end first. Poor low end response is the most common flaw of an inexpensive passive DI design using a cheap transformer.
Active DIs don't usually have low-end problems like inexpensive passives, or lose quality near the limits of the input signal, but a strong line-level signal like from a mixer or guitar with a preamp can overdrive the active circuit and very crappy-sounding distortion is the result. The inability to handle hot input signals is the most common flaw of inexpensive active DIs, which use lame power supplies.
Active DI performance also degrades as batteries drain, and sometimes if phantom power is not 48 volts. The ability to cleanly pass signal depends on available voltage.
Most of these shortcomings of both active and passive can be solved by using a good DI, except that you should probably use an active DI for passive instruments, unless you can try it out to see if it sounds OK.
Not all actives boost. Most don't. A DI that gives gain is really more of a DI with a preamp. Any DI circuit by itself actually drops the signal level. This lets you patch very hot signals into your board or recorder or whatever.