First, read a bit on meaning of word pulse.
The pulse current is the maximum current that battery can provide. If the current goes higher, battery can be damaged and will probably start overheating. The missing part is definition of the pulse. Somewhere in the datasheet it should be mentioned how long the pulse lasts. The other part which is also important is continuous current. It is the maximum sustained current which battery can provide and is usually much lower.
Cycle life I believe means the number of times the battery can be recharged.
One part you also missed are definitions of power, current and amount of electric charge. The charge a battery can store is measured in coulombs. It is shown in mAh because of the relation between those units. \$ampere=\frac{coulomb}{second}\$, so when you multiply ampere by second, you get the charge. the milli part is because traditionally small batteries have too low capacity to warrant use of Ah (which is used for lead-acid batteries). So to calculate roughly how long the battery will last, you need to divide the total capacity by the current you will draw from the battery. So if you're planning to use 190 mA, you'll get around \$\frac{85mAh}{190mA}\approx 0.44h\$. Do note that the battery will most likely not survive such use, because pulse current is usually higher than continuous current, which you'll need to take into account if you want to use battery with electrical motor.
As for resistor part, well the best idea would be to control the motor using variable voltage to set the current. So yes, you can use a resistor to limit current going through the motor. Do note that the resistor will use some of the power provided by the battery.
You are very close. The average power is a very accurate way to do this given that you are not pulling such a high current that the effective capacity of the battery fluctuates.
Batteries, Batteries, and More Batteries
There is one very important term, and that is the self discharge rate of the battery. This is dependent on chemistry, but lets say you get a nickel-metal hydride. The self discharge rate is
"20% or more in first 24 hours, plus 4% per day thereafter" if it is not a
low self discharge rate NiMH, which still discharges around 25 or so % a year.
Lithium batteries have some of the best characteristics for self discharge rate and my experience supports this fact. I think battery university has a great site to discuss many different battery characteristics and I often point people there to learn about batteries when they are starting to work with them. If you want to compare battery discharge rates they have an entire article discussing the phenomena.
This is a bit around the point, but I always try to make this point, when you measure battery voltage you need to have it under load. This varies with chemistry, but it is paramount in lithiums. I had a coworker placing bad coin cells in our devices and using them because the coin cells showed almost full voltage with no load. Under a load of any amount(10kohm aprox .2mA) they were flat dead.
Your Microcontroller and You
As you are dealing with using the manufacturer sheet on leakage current there are also many different issues you will have to deal with to keep to those specs that are probably also work thinking about. The biggest I have seen is a floating input. Many engineers will leave unused pins as inputs thinking, "Hey, what harm can this do?" Quite a bit if you are talking microamps. A floating input will have its transistors changing state constantly and the fluctuations cause a power draw difference. We once had a reduced lifetime in a product because we had an error that left 2 pins floating causing our standby current to more then double on our MSP430. You need to drive all of your pins to output and let them hold a state.
It is easy to miss when doing these calculations things like wakeup time. I seem to remember our MSP430 had a non-negligible wakeup time if you were doing it very often. It also had a larger power pulse for just a moment as it came online. Our little homespun RTOS had to try to take this into account and if the shutdown was less then X milliseconds we skipped it with NOPs and saved some power.
If you are looking at a very long life product, you are going to need conformal coating. The oils in your skin are not an issue immediately, but with time they form a lightly conductive material on your board. Conformal coating protects your board from this little current sucking side affect.
Read any app notes they have about low power operation, it probably covers issues like the pins need to be held as output and many other important and useful facts.
Last but not least, Dont let yourself get relaxed just because you have read the app notes and everything seems okay after a week of running your product, you have to do as clabacchio says, you must measure and make sure. You debug your code normally, this is part of it, you need to find out if you made a mistake that is causing your idle current to be mAs instead of uA or even just if you did what we did and a pin is floating on accident. Make sure you use buffered measurements when you do this, if you have a large leakage on your device taking the data you can make a mountain out of a molehill when testing. Also, never forget about pullups, they are little power hogs if you are not careful.
Best Answer
If you "forget about" internal resistance, then the maximum current is infinite. An "ideal" component, non-existent in the real world, can provide mathematically "pure" infinite or zero amounts of resistance, voltage, current, and all the rest.
Different battery compositions will have different amounts of real-world "impure" limitations. Internal resistance, temperature versus performance characteristics, "memory" and recovery effects, and so on.
One of the difficult times I had learning about electronics was doing calculations and then wondering why the physical components on the breadboard were different. The figures on paper say I should measure 9 volts. I'm actually measuring 8.654 volts. What gives?
A short length of wire might well be only 5 mΩ, but when you connect the battery using only the wire, it doesn't vaporize the wire with a massive surge of almost 2000 amperes. Why? Because the battery is limited by real-world physics.
Some batteries are capable of some extremely high current. Consider automotive "wet cell" lead batteries. You'll find that they're capable of 1000 amperes or more, especially for turning over huge engines during start. In electronics and physics, many things are a trade off. If you want super high current, you may have to accept lower voltage, lower battery life, or extremely high cost.
A capacitor, as another example, can supply extremely high currents (compared to batteries), but they store charge, and are not a charge pump, as a battery is. As such, they're sort of like super-high-speed batteries with extremely limited capacity.
It was the biggest eye-opener for me as a kid in school to realize that applying Ohm's law to components was not exactly straightforward. You have to take the physics into consideration, and it's messy. A capacitor isn't just a capacitor: it has some resistance and inductance as well. The best way to think about components and batteries, I think, is that any component is a mixture of a bunch of other components, but imagine a control panel with sliders. A resistor might have its "resistance" slider at a large amount, but the "capacitance" and "inductance" sliders can't be at zero. A wirewound resistor, for example, will have more inductance than say a carbon composition resistor.
Your math isn't wrong, but it's for ideal components. Check out a battery datasheet; it'll provide you with some figures that show where it isn't exactly ideal.
(If you happen to have a 2000A-capable 9V battery, I know some electric vehicle engineers that would like to chat with you!)