A motor's job is to turn with a certain speed and force. Up until the motor, the only power in a bot is electrical. The motor uses this electricity and some electric properties to spin an output shaft. So, the motor brings everything together and gives prior steps of propelling the bot meaning. They turn the electrical signals into force and speed.
Each motor is unique. Different motors have different properties, but all are governed by the same laws (ie. speed limit 65, slower traffic on right. Kidding, kidding, sorry). Each motor spins at a given speed and a given turning force at a given voltage. It also draws a given current from the batteries. To pick which motor you will use, you will need to know speed and force at a given voltage, as well as stall current. All of these you will learn below.
Take speed. Rotational speed is measured in RPM or revolutions per minute. This is simple enough: for 60 RPM, a motor shaft turns 60 times per minute, and takes 1/60 of a minute (a second) to turn all the way around. A great way to visualize this is a clock. The second hand completes one revolution for every minute. That's 1 RPM. The minute hand completes one revolution every hour, which is an RPH, or 1/60 RPM. The hour hand completes a revolution every 24 hours, or 1/24 RPH, or 1/1440 RPM. Needless to say, none of these three speeds are very usable with fighting bots, though.
But how can you tell whether a motor can power a bot? You need to know the turning force of the motor. The turning force is called torque. It's the force with which the shaft of the motor turns. This concept shows up in physics differently, with the equation: torque=force x distance. This would be, for example, the amount of force a kid places on a seesaw if he sits a given distance away from the pivot. But torque is different with motors: it is created by the motor. So, in the context of bots, torque is the amount of weight a motor shaft can support times the distance away it is. So, torque is measured in inch-pounds (or Newton-meters, foot-pounds, ounce-inches, etc: distance*length). The point is, as distance increases, force that the motor can supply decreases. For example, hold a ruler by the end in mid-air. Then add a weight halfway down. Your hand has to exert a torque so that the ruler doesn't rotate. Now move the weight all the way out to the end opposite to your hand. It just got even harder to keep the ruler still: twice as hard, to be exact. A motor is like your hand, and the weight of the bot is like.. the weight. The wheels section has more on this. To wrap torque up, here's a sample calculation: a motor has 2 inch-pounds of torque. It can support 2 lbs one inch away from the shaft. It can also support 1 lb 2 inches away from the shaft. And it can support 4 lbs a half-inch away and 1/2 lb (8oz) 4 inches away. Again, T=Fd (2*1, 1*2, 4*0.5, 0.5*4, etc =2).
The stall current is the current drawn when a motor is stalling. If you know the stall current at a certain voltage, you can find it at any voltage. This is because motors have a constant property called resistance (which will be discussed in Electronics), which is always equal to the voltage(in volts)/stall current(in amps). So, once you know stall current at a certain voltage, you can plug in your voltage and solve for stall current. Why is stall current important? Well, the greater the stall current, the more torque a motor has at a voltage (more on this in motor math). But, the ESCs must be able to handle the stall current since it passes through them on the way to the motors. Also, the batteries must be able to give the stall current, and the larger the stall current, the quicker the batteries are drained. Everything in fighting robotics has trade-offs.
That said, anything else you ought to know about motors? First, make sure you get motors that run on DC voltage, not AC (I think it defaults to DC if the power type is not mentioned). Unless you really know what you're doing, DC motors are simpler than AC, since batteries give direct current, and you can't plug a bot into your AC household outlet in mid-battle. Also, make sure the motor acts normally. If it has many leads and only moves a little when power is applied, you probably have a step motor. These are great for programmable robotics, but not for most radio-controlled fighting bots. And make sure the motor turns in one direction when the leads are connected to the batteries one way and the other direction when the leads are connected the other. This is how motors are reversed, and reversible motors are necessary for driving a fighting bot.
Otherwise, good luck and happy experimenting! Stay turned for Advanced Motor Math, coming fairly soon.