Weapon Types:
Brick/Box - Simplest of all bots, absolutely no weapons, all weight put into drivetrain and armor, so they are hard to destroy. Commonly boring bots, though, both to watch and drive.
Pusher - A bot which tries to control the match by outpowering the opponent in a pushing match. Since these require high levels of torque, they aren't usually very fast, and with no hazards they also are rarely stimulating to watch.
Rammer - A bot with no active weapons, but typically some sharp points or heavy weights to maximize damage. They attack the opponent by reaching a quick speed and ramming into them. Since there are no active weapons, more weight can be put into armor and drivetrain as well. Significantly more entertaining than brick bots and pushers.
Wedge - A bot with a sloped side so that it can get under opponents for greater pushing power, as well as disorient spinners by lifting them at an angle to the ground. The hard part about wedges is designing for the lowest ground clearance at the tip of the wedge. The tip of the wedge is almost always the part that catches the opponent, but if it is too high off the ground, it won't get under the opponent. If it's too low, it may scrape the ground and drag the bot, or otherwise hinder motion. Most people have a love-hate relationship with this bot type.
Lifter - This bot has an arm or some other lifting device designed to get under bots and lift them in the air, often disabling them as most bots' wheels don't always touch the ground. This design can then push around the bot until the driver is confident that the opponent will not be able to right itself. Probably the simplest active-weapon bot, and since speed is not a huge issue, motors can drive the arm instead of pneumatics. It also can right itself with its arm, if designed to. The only qualms with this bot are that most lifters aren't very interesting to watch - they only maneuver for the kill - and that not all bots are vulnerable to this kind of bot: invertible or well-designed bots which always land on their "feet" will not be very effectively beaten by lifters, most of the time. Also, frequently, the arm is vulnerable to powerful weapon-endowed bots. But lifters are a step up strategy-wise from wedges or rammers.
Clamp - This type of bot is a more advanced version of the lifter: in addition to lifting bots, this bot can clamp its prey, giving the well-designed version of this type the opportunity to carry triumphantly the helpless foe in its metal claws. I have never been beaten this way, but have lost in a helpless way before, and I expect this is the best way to really pi*s off the opponent (and I mean REALLY; otherwise I would have better censored myself). This bot doesn't seem like it would be very exciting (all it does is carry bots around if no hazards are at a competition), but there is a twisted kind of enjoyment the audience derives from watching a clamp in action, often. Although the opponent will probably not share in the fun! Often motor-powered, but the clamp may be a weak spot, and it may be difficult to line-up for a clamping. I think design is probably a large part of the success of a clamp bot.
Flipper - This type of bot is sometimes motorized, but lately has almost always been pneumatic-powered, using high-pressure gas let out at the driver's leisure to fire the cylinders which powerfully activate an arm whose goal is solely to flip the opponent on its side or back. Due to the sheer impulse of a flipper's arm going off, damage is often also done to the opponent during takeoff and landing, so to speak. This kind of bot involves a lot of strategy (air tanks can only hold so much air, so the cylinders can only be fired a certain number of times), and caution. It is dangerous to get near very powerful flippers when they are charged: that power could easily hurt someone in the arm's path. BE CAREFUL WITH THIS BOT TYPE! While it is one of the more exciting bot types out there, it is due to sheer power. Don't experiment with high-pressure pneumatics without some experience with lower pressure pneumatics, and even then, have adult supervision.
Thwackbot -
Vertical: Vertical thwackbots aren't very common, but they are still out there. This bot type uses motor torque and its own momentum to swing a weapon over itself to hit an opponent. These are probably hard to drive, though, as it takes constant reversing directions to fire the weapon. They are also hard to design, as it is hard to find a good, balanced design for the weapon part that does the most damage. However, these bots are interesting to watch, and so uncommon that a new one would probably stand out more than a new bot in a different bot design.
Horizontal: Horizontal thwackbots are the most common thwackbots, and are bots which often have weights or sharp objects at the end of a long rod. What makes a bot a "thwackbot" is that it spins in place, using this large extension as a weapon, similar to a baseball bat or pickaxe, to hit the opponents. This bot type is still considered a non-active weapon bot, probably since it is the bot itself moving, not the weapon. However, many successful bots have come from this category. The major weakness from this type is that it must sit in one place and spin, hoping the opponent will approach within weapon range. It would require a very short spin-up time for the bot to be fully effective if it came up to a bot and spun halfway around before hitting it. But still, this is a popular choice for a bot weapon type.
Spinners -
Vertical: A vertical spinner is a bot which spins a bar or disc (or ball-and chain.. any object, really) upward at a high speed, storing plenty of kinetic energy before impacting the opponent, often sending parts of the foe flying. Vertical spinners aren't as common as other types of spinners, partially because they are recent innovations, and partially due to dealing with gyroscopic effects. When a vertical spinner turns, the spinning disc does not want to change orientation (this is a physics concept called inertia), so the bot compensates by causing a turning force in the bot itself (I doubt I worded that right, though). Anyway, the point is, the bot tends to raise one side of the drivetrain while turning. This can be limited by turning slower, or by widening the base of the drivetrain, but it is definitely a weakness. Still, as bots of this nature gain effectiveness and destructiveness, there is little doubt that these bots will grow in popularity.
Horizontal: The most common type of spinner, a horizontal spinner spins a blade, disc, bar, etc. horizontally very quickly. This spinner may be located above, below or within the bot, with different effects. The point is, this is one of the most common bots out there, mainly because they are among the most destructive. When a blade spins very quickly, it has plenty of angular velocity to lose if it is hit, so it is harder to stop. Also the blade itself has a certain measure of how difficult it is to start and stop (again, inertia) which is factored in with speed to get kinetic energy. When kinetic energy is expended, really what is happening is the opponent has found its way to your spinning object, and the spinning object does not want to stop spinning. It resists it, and the more it resists it, the more damage it will do to the opponent (which is in the way of its spin path). The more KE, the more destruction. Spinners thrive on this. Their main weaknesses are wedges and their annoying habits of disorienting spinning weapons, as well as spinners themselves: spinners receive as much trauma as they dish out, and often disable themselves if they aren't designed right.
Shell: This spinner is interesting: it spins an outer shell to a high speed, protecting the innards with the weapon, and gaining inertia with the shell shape. However, these spinners frequently are not invertible, and if the spinner dies, they have little offense (and sometimes little defense). It's also hard to tell which direction is forward, especially in a hit where both bots are sent flying and spinning. But I must admit, these are really fun to watch.
Drumbots - A twist on a conventional spinner, drumbots are rapidly gaining popularity. Their weapon is a cylinder with extensions or indentations (or generally sharp, pointy things) which spins at a high speed and hits other bots. This kind of bot has a quicker spinup than usual yet still packs kinetic energy, and frequently can be invertible, hard-hitting and frequently-hitting. However, wedges have a nasty habit of outdoing this kind of bot as well, getting under the drumbot and driving it around the arena. Also, someone brought up the point that shrapnel could get stuck between the drum and the bot, stopping the weapon and stalling the motors, another good point. It would have to be strong shrapnel, but it's still a possibility.
Hammerbots - These bots use pneumatics or motors to propel a hammer, be it blunt or sharp, in order to hit the other enemy, possibly causing a dent or even a puncture in the opponent's armor. At times, if the armor is punctured, the hammer can even impact some vital part of the bot, causing a knockout. These bots are most effective against bots with weak top armor, and won't necessarily punch a hole in every material. They also require accurate aim to fire the weapon. However, they can be fun to watch, and effective if properly designed. They are on the decrease, but there is still room for hammerbots: with hammers on the decrease, people are worrying less about their top armor, leaving an opening for a return!
Spearbots - These bots use stationary or pneumatic-powered spikes to ram into opponents, in hopes that the spike may puncture through the foe's armor, possibly disabling the opponent. With a pneumatic spike, there is an extra jolt on impact, and with accurate timing, if the bot impacts as the spear fires, more damage is usually caused. However, these bots aren't frequently effective, and aren't common either.
Crushers - These bots use hydraulic systems to slowly but powerfully exert pressure on a fine point when the opponent is in the bot's jaws, often puncturing the armor. These crushers are effective due to the small point of contact on the pincer, as well as the sheer power of hydraulics: hydraulics are much slower than pneumatics, usually, but because of that slowness much higher pressures can be obtained, meaning that the water which powers the system can push the cylinder much harder, and therefore the pincer is compressed harder. However, hydraulics systems are heavy: it usually is confined to middleweights (120 lbs) or heavier (and I expect more times than not, heavier) due to the sheer weight the pumping system adds. It is also much more dangerous than pneumatics. I read somewhere that a hole in a hydraulic system could easily blow off an arm or leg with the resulting stream of water, or forcefully inject water or oil into the body. Not pleasant stuff. I seriously recommend that you not try this at all unless you work with hydraulics all the time: a bot is no reason to lose your right leg. But admittably, crushers are often successful. The danger factor and weight keeps most people away from them, though.
So what is the best weapon type!? It depends on the person. Everyone has different preferences. Some of the best bots, though, are combinations of weapon types (such as wedge and spinner, for example, in the greatly successful Hazard) or completely new ideas, sometimes revolutionizing the playing field (I think, for BattleBots 1999, Jim Smentowski's vertical spinner Nightmare was the first vertical spinner ever made, now there are lots of new vertical spinners). So, don't be afraid to try something different. But at the same time, as the saying goes, K.I.S.S.: Keep It Simple, Stupid. Don't overdo new technologies. The more things you are experimenting with at once, the more failure points there are. Try to simplify as much as your design allows. You may not directly be glad for it (it may fail some other way), but you may be sorry if you don't. Good luck!