We use them because they have a very high power output for their relatively small volume and weight.
You must understand that puncturing or shorting a Lithium Polymer Battery is arguably the worst accident that can happen in combat robotics.
Protect your batteries. Respect your batteries. Failure to do so can and will jeopardize the health of everyone in the building where it happens.
When punctured, they explode or catch fire, releasing a slurry of harmful solids, liquids, and gases, including aerosolized lithium and highly toxic hydrogen flouride gas.
This isn't something that can just be extinguished and cleaned up, this would render any area exposed unusable until hazmat deals with it.
You'll find these on your battery pack:
Capacity in mAh (supplies x milliamps * hours)
C-rating (can supply x amps at any given time)
Voltage in V ( ͡~ ͜ʖ ͡°)
<x>S (means the battery has "x" cells)
The marked voltage on your battery is actually the lowest voltage that you should ever let your battery reach. LiPo cells must not be discharged beyond this point or else they may be damaged.
Your robot isn't going to just shut off when you reach the minimum safe voltage for your battery. You'll likely have to estimate a good capacity battery to use based on your electronics, then test how much the voltage drops over about 3 minutes of standard use. Ideally, you want your battery to be almost fully spent by the end of the match, to minimize the amount of excess lithium you're lugging around.
Some people will tell you to charge your batteries as fast as you can. I am rarely in a rush to get my battery charged. I like to charge my batteries at 1C, meaning that they'll barely get hot. You can do this by taking the capacity of the battery, divide by 1 hour, and charge it at this many amps. Consequently, whenever someone asks me how long it takes to charge, I can say " 'bout an hour."
It is absolutely imperative that you use the balance plug when charging LiPo batteries; the charger itself may not stop you from charging them without it. This ensures that the same amount of charge is imparted on each cell, such that the charger doesn't accidentally charge one of the 3.7V cells up to 12.6V alone (that would cause a fire / explosion)
Do not make your own charging cables, leave that to leadership.
Plug the charger into the wall. The screen will turn on.
Plug the adapter into the charger. The adapter comprises two wires, going into one end that matches the battery, and one that goes into the little banana plugs on the charger. Do not connect the adapter to the battery without if the adapter is not connected to the charger first.
Plug the balance plug (the smaller plug with [battery's cell count] + 1 pins) into the charger.
Plug the power plug (the bigger plug with only a red and black wire on it) into the adapter.
Click the left and right buttons on the charger until the program at the top reads "LIPO CHARGE"
Click ENTER, and the current will start flashing. Read the capacity of your battery in mAh (it's written on the battery). Convert to Ah (divide by 1000). This is the current at which you should charge your battery. Set this using the left and right buttons.
Click ENTER, the voltage will start flashing. Click left and right until this number matches the one on your battery.
Double check both of the numbers you set. Press and hold ENTER.
The charger will effectively ask you, "are you sure?" press ENTER again if the information on the screen looks correct.
Once you see a timer start, the battery is charging. Note the voltage displayed in the top right of the screen.
It's not ideal to store your batteries with a very high or low charge, nominally, you store them around 80%. Since there is an (effectively) direct relationship between the voltage across the terminals of the battery and the charge level, you can figure out a good level to leave it at. If anyone actually reads this, ping me and remind me to add a table of nominal storage voltages and how to use the LIPO STORAGE program on the chargers ;).
I will preface this with a TL;DR...
Most robots in our club use a 14.8V, 650mAh or 850mAh, 100C battery. If you're doing something conventional, this will probably be just fine for you.
You'll either use 11.1V or 14.8V LiPo's for most conventional robots. The capacity and discharge rate (C-rating) are arguably far more important.
Take the sum of the max current drawn by every component in your robot (call it I_net). You'll want a C-Rating a little higher than this.
multiply 3/60 (the amount of an hour you need your bot to run for) and multiply this by I_net and that is the "battery capacity you would need to run everything on your robot at the same time at maximum power for the entire match." Our bot Liberator's battery capacity was ~70% of this and that was arguably far too much. You'll want to gauge the size of your battery with reference to how you plan to drive it. I apologize that there isn't a cleaner cut way to measure this.