I am desinging some experiment using a 9Ahr, 12 V as the power supply. I am not sure how many percents of the energy the battery can supply. For example if my experiment consumes a current of 0.1A, then how many hours the battery can be used?
The "9 Ah" rating should be a measure of the useful power you can extract from it not some theoretical maximum.
The rating is usually based on a discharge time of 1 hour, i.e. if it is fully charged the battery will deliver 9A for 1 hour.
There is not likely to be any problem with currents smaller than that, so it will deliver 0.1A for 90 hours.
For very small currents, you also need to take account of the self-discharge rate of the battery (i.e. how fast it goes flat when it is not connected to a circuit) but that is only relevant if the expected battery life is months or years, not a few days as in your case.
Get hold of the data sheet for the Battery. Whilst what AlephZero says is generally true that a 9AHr battery will give 90hrs at 0.1A your particular application may give different results.
The AH rating of the cell is specified at a fixed discharge current "C", the AH are generally constant when the actual discharge is close to "C", but if you are operating at 10 or 20 times "C" you get much less capacity from the cell.
The Battery data sheet will give a number of graphs that show voltage drop off over time at various ratings of "C". You need to check your minimum operating voltage against time for your drawn value of "C".
If you are going to use this device regularly and want to keep the battery healthy, its best to calculate the theoretical time that the battery can produce a useful output, then divide your number by 2 or 3. Do not exceed this time for regular useage. The reason why is because you are using a lead acid battery. It is a fact that lead acid batteries will die much faster if you do deep discharges vs shallow discharges.
Look for 'deep discharge' or 'marine use' batteries rather than regular car/truck type. These are designed to allow you deplete them by upto 90% without damage, most car/truck types can only be discharged to around 50%, although they can supply much higher current for short periods - needed for starter motors.