1) LED's do not have a mAh rating. Ampere Hours (or miliAmpereHours are capacity ratings for storage cells, or Batteries. It means that a particular unit can deliver n mA over a one hour period. That is NOT to say that if you were to short the + & - that the current would top out at n mA. It would, more likely, deliver several Amperes instantaneously and until the delivery mechanics of the battery (chemical reaction, charge depletion, whatever) were overcome by the rapid discharge, or the short melted, ect. A 200AH 9V battery would be one hellashious honker. That must be a missprint on somebody's part. 200mAH seems a little low.
2) Toke gave you a very good, very basic, lesson in series circuit Ohm's Law. You have a series DC circuit (single current path, Direct Current) supplied by 9 Volt source. The total voltage drop accross that circuit will be 9 Volts, some dropped by each circuit element ( resistor and LED) . The Absolute max ratings for your diode are Vf=3Volts at 20mA (MAXIMUM). That says that under MAX conditiond, the LED will drop 3Volts, so the resistor has to drop the other 6Volts. Ohm's law tells us that in a series circuit, each element will pass the same current (20mA at the most). So the resistor has to drop 6Volts (V=6) and pass .02A (I= 20mA= 0.020A). Ohm's Law says that Resistance (R in Ohms) is equal to Voltage (V in Volts) devided by Amperage (I in Amps) hense, R=E/I= 6/0.02= 300 Ohms to avoid exceeding the max ratings of the diode. I would dissagree with Toke, in that I would use a 330 Ohm instead of the 270 to stay within the limits, but with resistor tollerances and manufacturer's tendancy to overstate Absolute Max's, you'll probablly be just fine using a 270 Ohm resistor. It don't get any simpler than that, bud. If you're still confused, take Toke's advice and stay away from DIY electronics. But, Good Luck with your project anyway. Don't parallel the LED's! You'll end up frying one of them, unless you really meant what you said about not caring if you decreased the diode's life (if that is the case, you can get maximum brightness, albeit for a very short time span, by just tying it into the battery without any current limiters). Well, that's about as much yabbeling as I can do. Back to you Toke.