Yes, it can be done, with due care.
The laptop battery described could typically provide quite a few jump starts. Getting it to do so safely and effectively is "the trick". Read on ...
What is often not generally appreciated is that energy to turn the starter motor during a jump start does NOT come directly from jump-battery to starter but is first transferred from jump-battery to car-battery and then from car-battery to starter motor. (See note at end)
The starter motor draws hundreds of amps - probably 100-200 Amps in a very good situation and 400-600-800 Amps in various other cases. Even when the car-battery has been run flat by lights left on or whatever, it has the ability to accept energy from the external source and then to return it to the starter motor.
A 11.1 V LiIon battery probably has 3 cells in series.
11.1/3 = 3.7V/cell. Actual cell voltages will be about 4.2V fully charged and 3V dead dead, so the battery = 9V to 12.6V. It is OK to use down to about 9.3V. At 9V if it does actually charge you are in the area where it is doing damage to the battery to use it.
Issues are
If car battery is very flat its terminal voltage may be low enough that the laptop-battery provides more current than is wise. The laptop-battery can be damaged and lose cycle capacity and worst case can "vent with flame" - spectacular high energy melt down.
If the car battery is very high impedance due to eg major internal sulphation the voltage may be around 12V when open circuit. The laptop battery open circuit voltage may be lower than the car-battery voltage and no energy transfer may occur. This will probably not usually be what happens.
If the laptop battery was rather flat (around 9V) and the car battery had modest life in it (eg starter relay goes click-click-click but starter will not turn, headlights light dimly) then the car-battery voltage may be usefully greater than the laptop-battery voltage and the charging may occur in the opposite direction. Max safe charging voltage for your laptop battery is probably about 8.4A (A = Ah rate). A battery that may not start a car may happily deliver 10 x as much current as the laptop battery can safely handle.
"Vent with flames" / fireball / pyrotechnics may well eventuate. you'd be needing a new laptop battery. The car battery would not have noticed.
In many cases you may be able to 'just connect the batteries " +ve to +ve, -ve to -ve and get an acceptable result. But, as above, maybe not.
Adding a series diode of suitable current rating will protect against charging the laptop battery. You can still get excessive current the other way. And voltage available is reduced.
The laptop battery would be good for this purpose if properly used.
What is needed is some limit on safe maximum current, and a voltage for charging that is suited to the car battery in any state. You could make such but you can but them on ebay for less than you can make them for - or from China for perhaps slightly less again.
What you need is a power supply with Vin is at least 9V - 13V, Vout is say 13V-14V and current out is limited to say 5A. If the car battery loads down the supply then the current limit should work. such supplies probably cost in the $US5-10 range on ebay. You can pay more.
Note: You will find much discussion on whether jumpstart energy is first saved in the car battery. I'm wholly convinced that it is. I have started cars from battery packs joined with twisted connections and thin wire and connected to the battery by 3 feet + of "lamp cord.
Eureka! Very satisfying when it works. A starter motor would not even notice that such a battery was there.
If you have REALLY heavy jump start leads, big solid connecting clips and some luck then you may be able to jump start directly from the remote battery. I've seen it done once when all normal attempts failed. The AA was called and he had a monster battery and lead set. It only just worked and his leads nearly melted. It charged and ran OK after that and I have no idea why it behaved that way. ALL the jumps start packs sold have no chance of starting a car directly.
Since you're new to EE, I'm going to have to say that the answer is no. What you are asking is pretty straightforward, but it's not something you can do with hardware-store parts.
Basically, what you need is an energy storage device that you can insert between your car battery and your computer. This is complicated by the fact that you need to isolate this storage from variations in the battery voltage. That is, when you try to start the car and battery voltage drops, you don't want your storage device to try to backfeed the battery, as this will exhaust the storage device and your computer will die. In principle, you can do this one of two ways:
simulate this circuit – Schematic created using CircuitLab
Note that I'm showing your storage device as BOTH a capacitor and a battery. This is not necessary - you can use either. Using a battery instead of a capacitor is in some ways easier, but you have to worry about keeping it charged, which can be a royal pain in the whatzit. On the other hand, you don't have to worry about making sure the capacitor is charged before starting the car, which can be a problem, too.
Also note that the second approach, storing energy after the converter, seems simpler because you don't need a diode to keep from backfeeding the car battery, but this may or may not be true, depending on the converter. It assumes that the converter will not let current flow from output to input when the converter is not working. This may or may not be true. If you select a battery, it also assumes the converter will turn on properly with a humongous capacitor hanging off its output. This also may or may not be true.
Using a capacitor will work, but you have to size the capacitor properly. To do that you have to know a) the current it supplies when needed, b) how long it needs to provide the current, and c) what voltage drop is acceptable during that period. If the answers are a) i amps, b) t seconds, and c) DV volts, then
C = 1,000,000 x i x t / DV (in uF)
So, if you're supply 2 amps for 3 seconds, and you can accept a voltage drop of 2 volts,
C = 1,000,000 x 2 x 3 / 2
= 3,000,000 uf, or 3 F.
You'll have trouble finding a 3 F, 12 or 24 volt capacitor.
So, like I say, this may be rather more trouble than you can handle at this stage in your education.
Best Answer
Usually it's better to leave the old battery in there as it can still help with starting the car even if it can't manage on its own. Inexpensive thin jumper cables or jumper cables connected to sub-optimal connection points can easily have too much voltage drop for the starter motor to be able turn over an engine. Sometimes you can let the old battery charge for some time (maybe 30-60 minutes) by connecting it with a running car or using an external charger and that will be enough to get it moving.
The problems are worse again in winter when batteries have reduced current capacity and oils are thick. High quality jumper cables can make all the difference. You'll know there is a voltage drop problem when the lights on the dash just about go out when the engine is supposed to be cranking. Maybe the starter just clicks. If the lights don't dim and the engine isn't turning over it could be the starter itself. There are only a few things in the circuit and it's pretty easy to troubleshoot (but if the starter is shot it can be a job better suited to a pro to replace it- some of them are very hard to access without a hoist).
However, I did spend a good part of an evening with friends trying to jump start a small crossover SUV without success, the motor just wouldn't turn over even with a freshly charged brand-new battery and high quality connections. Turned out that the engine had seized internally (it was apparently thunking lustily when shut off) and the vehicle was practically a write-off.