Much of this information is in the High-Powered Lighting Systems article, but I've put the info here in a somewhat more informal way.
The most powerful lights, and the most expensive ones, are those with separate lighting heads and batteries. The battery is a substantial cost of the lighting system. The battery is also the most likely thing to die and leave your lighting system unusable. So taking care of your battery is essential. Unfortunately, many systems come with lousy chargers and inadequate instructions, and there are quite a few myths out there about rechargeable batteries that make things even more difficult.
The three major kinds of battery are sealed lead-acid, NiCad (nickel cadmium), and NiMH (nickel-metal hydride). As you go from lead-acid to NiCad to NiMH, the batteries get more expensive, and are smaller and lighter for the same capacity. So if you have lots of money you probably have a NiMH system, while if you're on a tight budget you probably have one with a lead-acid battery. These batteries have different characterists and charging requirements, but they all have a couple things in common: none of these batteries likes to be overcharged, and you should never 100% discharge any battery. So when your lights go dim and yellow, no matter what kind of battery you have, turn off your lights.
Lead-acid batteries slowly drop in voltage as they power your lights. This means that your lights will dim slowly as they get empty. Once they get noticeably dim, turn them off, as lead-acid batteries are easily damaged when discharged too low. They are not very good at providing high currents: if you hook them up to a very bright bulb they'll go flat very quickly. What's too high a current for your lead-acid battery depends on the voltage of the battery (whether 6 or 12V) and the make of battery. For a standard cheap 6V battery, 10W is about the most you can get out of it, while a Hawker Cyclon battery can supply 20W easily. If you want more than this, you'll have to use NiCads or NiMH batteries. Lead-acid batteries also lose capacity when it's cold (near or below freezing).
Lead-acid batteries are best charged using a constant voltage source. There are two basic ways of charging them: quickly and slowly. The voltage for a quick charge (also called bulk charge) is higher than the voltage for the slow charge (also called float charge). The problem is that if you leave your battery attached to the higher (bulk) voltage for too long, you will destroy it. So a proper fast charger detects when the battery is full and switches to a float voltage. If you attach your battery to the float voltage, it charges slowly, but then you can leave it hooked up to this voltage forever and the battery will be just fine.
The chargers that come with lighting systems that use lead-acid batteries are almost always simple AC/DC chargers. They usually apply a voltage that is far too high for the battery, and will kill it if you leave the battery plugged in for too long. Unfortunately, while the instructions you get with the lights tell you how long a full charge would be (i.e., how long you'd charge after you drained the battery until the lights went noticeably dim), if you recharge for this entire amount a battery that's only partly discharged you will end up overcharging it. So you have to guess how much of the capacity you've used, and cut down the charging accordingly. So for example, if you estimate that you've used half the capacity of the battery and 8 hours is a full charge, then charge for at most 4 hours. But this requires guessing how much you've used the light, which can be difficult if you're using different powers of bulbs or turning your lights off when you're not using them.
To avoid this guesswork and keep your battery fully charged but not overcharged, you should get a proper charger. You can buy a good fast charger from Maplin (see their catalog, look left for headings Electrical, then Batteries & Chargers, then Chargers then Sealed Lead Acid Battery Chargers, they cost 30 quid or more), or you can make your own float charger for very little money (see my article on it). While it may seem like alot of expense or bother for a lighting system that's reasonably cheap, remember that it won't work at all if your battery dies through having been overcharged or over-discharged.
Lead-acid batteries last the longest when they are kept topped up, so you should charge them as soon as possible after you use them, and when you put them away for the summer you should make sure they're fully charged up.
NiCads are a better choice than lead-acid batteries for many situations. They stay at about the same voltage through most of their discharge cycle, so they keep your lights nice and bright until the battery is nearly empty, then they go dim very quickly. They are generally a fair bit lighter than a lead-acid battery for the same capacity, and often smaller as well. They can handle high currents better than lead-acid batteries. They also work better in the cold, although they are affected by extreme cold (like you get in Alaska). However, they self-discharge, losing 1-3% of their capacity per day, so they aren't as useful if you want to have a rechargeable system ready to go at a moment's notice.
Like lead-acid batteries, they suffer when overcharged and can be damaged by being fully discharged. Unfortunately, there seem to be quite a few myths about these batteries. The most prevalent one is that they can't be topped up after being only partly discharged because they have "memory", so they will remember this and only allow you to use this amount of capacity in the future.
This only bears a faint resemblance to the facts. NiCads do suffer from "memory" but it is not a reduction of the capacity of the battery, but a reduction of the voltage of the battery. A battery suffering from memory will still power your lights, but since it does so at a slightly lower voltage than it would if it were in top condition, the lights look a bit dim and yellowish. Second, it is not caused by topping up your battery after only a partial discharge; it is caused by overcharging your battery by leaving it plugged into a stupid charger (the kind supplied with most bike lighting systems) after the battery is already charged. Third, memory isn't permanent. If you have a battery suffering from memory (dim lights even when fully charged), here's how you fix it: do a carefully controlled discharge. What you want to do is discharge the battery until its voltage is down to 1V/cell. For a 6V battery, this is 5V. If you don't have a voltmeter, you can approximate this by discharging until your lights start to get noticeably dim and yellow. Then turn the lights off: never 100% discharge your battery, or you can destroy it.
The best way to avoid getting memory is using a smart charger. This charges your battery until it is full, then switches to a tiny current that counteracts self-discharge. Smart chargers for NiCads batteries are harder to find than for lead-acid, so the easiest route is to modify a power tool charger (info here). If you don't want the expense or trouble of this you should try to estimate how much you've discharged your battery, and cut down the charge time accordingly, as described for lead-acid batteries. Then if you discover you've got memory, do the carefully controlled discharge described above.
NiCads don't need to be stored full; they can be left in whatever state they were when you finished your ride. Since they self-discharge, I tend to leave them uncharged until the evening before my next night ride, when I charge them up.
More bike and light articles
NiMH batteries are the best technology currently offered in bike lights, and they are correspondingly the most expensive. They tend to be smaller and lighter than NiCads with a similar capacity. In many ways they are similar to NiCads: they self discharge at about the same rate, they can supply high currents well, they work better than lead-acid batteries in low temperatures, and they can be left in any state (charged or part discharged) when stored.
Like NiCads they suffer from overcharging. There is a myth that since NiMHs don't suffer from memory that you can charge them up as often or and as much as you like. This very misleading: NiMHs don't suffer from "memory", but they dislike overcharging even more than NiCads. The result of overcharging is permanent damage to the battery rather than the curable problem of "memory". If you overcharge your battery using a stupid charger, you will greatly reduce its lifetime. This is why many systems come with smart chargers. If you have a choice between similar systems, one with a stupid charger and one with a smart charger, go for the second one. It will save you buying a new battery if you kill the first one.