This is the driver found in my Ultrafire C12. Is this a direct-drive type of driver? Or a constant current type? I noticed the light output is very strong at the beginning but gradually become less as the battery depletes.
Are R1 and R2 sense resistors? If I added a smaller resistor across them, will I be getting a bigger current to the LED?
Thanks in advance for helping to clarify my doubts! 
It is direct drive.
With a single cell and moderately high drive current only a boost driver will maintain constant light output as battery voltage falls (as long as the battery is able to deliver the increasing current to the driver as voltage falls). Linear or 'constant current' will all drop in light output & current when the input voltage falls below the voltage needed to put the set current thru the LED. The only exception is if you have a LED with very low Vf, or a very low drive current (where the LED's Vf is also very low).
Many thanks!
Is there a way to increase the current to the LED with this driver?
Use a better cell and/or figure out what FET it's using and replace with a lower resistance part. I don't see any limiting resistors on there.
edit: unless there's important parts hiding on the side of the board hidden by the wires. Better pics would help.
There are no other parts hidden by the wires. There are 2 resistors, labeled R1 and R2. There is a ATMEL-240-24C02BN (Serial EEPROM?) but the other IC is unlabeled.
The black LED- wire is attached to one leg of a 3-pin component. Are there any markings on that one?
The only resistors visible are tiny 0603 size, those aren't physically capable of being used as limiting resistors.