Li-ion battery DC voltage

mA-hour rating is not important here - the larger the better for you. It is a measure of how long the battery lasts. What you need to do is make sure the batteries supply the current at the same potential. Your present set up has seven 1.2 Vdc cells in series: that's 8.4 Vdc, whereas your Li-ion battery only delivers 7.2 Volts. The Li-ion battery won't run the motor as fast as your present set-up, and it will stall out easier, too. However, should the 7.2 Vdc battery suffice, it should run roughly twice as long as the 1500 mA-hour batteries you have right now.

I agree with EMC_C. The mAh is a measure of how big the tank is. A small motor might draw 1000mA @ 7.2V. If it draws 1000mA, the 1500mAh battery will last 1500/1000 = 1.5 hours. The 3000mAh battery will last three hours.

You probably want some kind power supply between your motor and battery so that the voltage to the motor stays within its acceptable range and so the battery isn't over-drained. You can damage a battery by trying to draw current from it after its voltage has fallen below a certain level.

LI-Ion batteries are fairly common in model aircraft nowdays (I am old, we always used internal combustion engines) you can get all the info you ever wanted to know by visiting the myriad web sites devoted to model aviation.

Get an L-Ion at somewhere around 18v and put in a voltage control circuit to back it down to 8.4 volts at the power rating the motor is subjected too. This will give you a higher power density without spending your childrens inheritence.

And I want to upgrade the operation of my gadget. I have right now an Li-ion Battery pack (7.2Vdc; 3000mAh). But I'm afraid to use this type of battery, maybe it will spoil the gears as well the dynamu.

No need to worry. The larger capacity battery (3000 mAh) will simply allow the circuit to run longer. It is the difference in voltage that could change the performance of your motor, i.e, make it run slightly slower, if there is no control over voltage and current. (It is possible, if there is a motor control circuit involved, that the maximum motor voltage is, for example, 6 volts -- in which case there would be no reduction in speed with the lower battery voltage.)

It would be great if maybe you guys can explain the purpose of a mAh of a battery

The purpose of the mAh rating (or for larger batteries, the Ah rating) is to tell the designer or user how long the battery can be used to supply a load before it needs to be recharged. If this were a perfect world, your battery could supply a 3000 mA load for one hour, a 1500 mA load for two hours, a 1000 mA load for 3 hours, etc. In fact, batteries do not work in precisely that way, but for many purposes, that principal is close enough. (In some batteries, notably lead acid, the Ah rating can change dramatically, depending upon the test load.)

So normally, there would be no need to reduce 3000 mAh to 1500 mAh, unless you are concerned about cost, and not concerned about how long the device will operate between charges. If the motor is contolled only by an on-off switch, the reduction in speed (original speed x 7.2/8.4) will not damage the motor or anything attached to it, unless the device is right at the limit in terms of motor cooling, in which case lower speed might cause the motor to overheat.

If the motor has a speed control, it would typically be pulse width modulated ("chopper") and will function about the same on the slightly lower voltage, again, unless the original design is too close to the limit. (Going in the opposite direction is also usually not a problem: in the taction motor world, for instance, a controller nominally rated for 48 volts will be able to accept voltages up to 60 volts on a normal operating basis, and will often be able to accept voltages at a comfortable margin above that, to allow for normal production and usage variations.)

So the short answer: Your device will operate at a lower rpm, (85% of original) but will operate for about twice as long.

First, take one of your Ni-Cad batteries out of the circuit to make sure your gadget works on 7.2 volts. If this is OK then you have no problems with your 7.2v higher capacity battery.

Good suggestion.