I've been studying the Phillipino martial art Kali for about 6 months now, in combination with western boxing, Thai kickboxing, and grappling - all as part of jeet kune do, which was created by Bruce Lee in the '60s.
I'm far from being any sort of authority on this, but the information available on the web is pretty limited, so I thought I'd add my 2p-worth.
The word "kali" is an amalgam of two Phillipino words meaning "the study of body movement". It encompasses weapons-based fighting as well as "empty hand". The weapons I've used so far have predominantly been the rattan stick, either single or double, and the knife, and combinations thereof. I've also tried some staff fighting. The staff comes under the 7th (or 8th?) area of Kali. I'm not sure of the areas, but I presume the stick and knife stuff is in the early areas. Later areas also include using an oar, swords, and throwing weapons. Kali can also be referred to as arnis, eskrima, or Pencak Silat, although I suspect there are clear distinctions here that I'm unaware of.
My teacher uses the stick training to augment body movement, and so far this is where I have found it most useful. If you take the stick out of your hand and try the stick drills empty-handed, you realise that you're actually doing jab, hook, uppercut and back-fist punches, more or less. Other teachers have a different focus, with some concentrating on hwo to actually fight with sticks. While this may have some bearing in some societies, and of course in competition, I still find the stick most useful to train myself into using correct footwork, and getting used to right or left hand lead stances.
Kali employs what is known as "triangular footwork". This basically means that you stand at the base or point of an imaginary triagle on the floor. If your feet are together, you are standing at a point - if you step forward and outward with your right foot, for example, you can imagine that your left foot stays on point 1, and your right foot is now on point 2. If you move your right foot back to meet your left, then step out and forward on the left foot, your left foot is now on the 3rd point of the triangle. If you start with your feet apart, they are on 2 points of the triangle already. The triangle can be pointing forwards (called "male triangle") or backwards (called "female triangle"). This all sounds overly complex, but in practice it's very simple. Any boxer will be familiar with this footwork, it's just a naming convention.
Working right lead and left lead is a part of Kali I find quite tricky. Normal boxing stance is left hand and leg forward, for a right-handed person. Kali requires you to train "southpaw" as well as in conventional stance, i.e. with your right hand and right leg forward, if you are right-handed.
This use of both conventional and southpaw stance helps you to understand another of Kali's concepts: angles of attack. A left handed person throwing a jab, or a right handed person throwing a cross will land on your chin from eactly the same angle. Kali teaches you that you can respond in eactly the same way to these two apparently different punches - in theory anyway. Of course, a cross is usually very hard, and a jab usually very fast, so in practice, some of the evasions/blocks/etc are more useful than others. This is my experience: your mileage may vary of course.
Another ethos my teachers have stressed is that in Kali you attack the weapon, not the man. I'm not sure how true this is outside of stick fighting. Stick fighting, in my understanding, grew as a result of secret sword training during Spanish occupation, when carrying of swords was banned. Stick is taught at three ranges: largo (long - you can hit the stick and the opponent's hand); medio (medium - you can hit your opponent's stick, and pretty much anywhere on their body); and corto (short - this is grappling and trapping range, and punches, headbutts etc come into play). Consequently, a stick fighter will look to stay at long range, and strike the opponent's hand to make them drop the stick. This is attacking the weapon - my teacher sees this as a humane alternative to getting in close and striking the head - and of course you hopefully avoid the danger yourself.
In the empty-hand style, attacking the weapon becomes far more unpleasant. The use of elbows to block an opponent's punch means that they should end up with broken knuckles. At mid-range (i.e. boxing range), Kali can be used to perform wrenches on arms, and some punches are delivered into sensitive parts of the arm, hopefully to render it far less effective. I've been taught never to go looking for a limb destruction, but to consider them "incidental or accidental" to a fight. Ditto for trapping. I like this attitude, it seems very realistic to me. Limb destruction and trapping don't seem like useful techniques in isolation, but they can definitely augment an effective boxer's repertoire.
Attacking the weapon in knife-based Kali is of course far more potent. As your opponent slashes at you, you can influence the path of their hand (if you're damn quick), avoid the strike, slice into their arm, and potentially disarm them in the process. (Possible nonsense: I suspect that the crazy knife fight in The Bourne Identity involved a lot of Kali choreography. Matt Damon stabs his assailant in the arms repeatedly with a biro.) Kali is quite surgical here - stab or slice a forearm muscle and it can't contract, leading to your opponent hopefully dropping the knife.
There are a few similarities that I have noticed between Bruce Lee's teachings, and some elements of Kali. "attacking the weapon" seems to me to be similar to "cutting the tool" - this is an incidental strike to a limb during a punch. Imagine your opponent throws a straight cross - you slip it to the outside, and punch over their arm into their chin. Your forearm will "cut into" their bicep. My interpretation here may be quite naive, though. Also, Bruce Lee notoriously trained southpaw, to put his most effective weapons closest to his opponent. Kali's emphasis on using both leads equally seems to me to be a similar notion, but it makes both sides of your body effective, rather than admitting one side is better than the other.
Kali footwork also involves what is known as "sectoring" or "zoning". This, very simply, means stepping to the side of your opponent, or into their stance (i.e. between their legs), standing on their foot, or getting behind them. The footwork that puts you into these positions is, for me, tricky to get exactly right during sparring. However it is extremely effective for take-downs, and unbalancing your opponent. Sectoring could be seen as simply "getting out of the way", but Kali uses it both for evasion, and aggressively, to shove an attacker to the floor. It's very difficult to do this during training, since the falls can be very awkward. They are always followed up with limb destruction (e.g. a stamp on the knee/leg/ankle) or a strike to the head/body.
Recently, I attended a seminar by Dan Inosanto, one of Bruce Lee's training partners. This man's depth of knowledge is truly formidable. He told stories of how Phillipinos of his father's age enjoyed entering boxing contests staged by USA troops in the Phillipines. From their point of view, western boxing is almost "easy" compared to Kali - no tripping, no kicking, no backfist punching, no headbutts, no elbows... My teacher often refers to Kali as "boxing with all the dirty tricks kept in" - a very accurate description.
My teachers use elements of Brazilian Jiu-jitsu and vale tudo for take-downs and grappling. Aspects of original jeet kune do grappling are also taught, although the pervading opinion seems to be that recent innovations in martial arts have improved on these.