You car battery, or the connections to it, are bad. It can run on the alternator once engine is going.
sounds like a bad alternator. Take vehicle to a parts store like Autozone and ask them to test it!
If you use the radio without the engine running you will eventually discharge the power from the battery. It should be fine for a couple of hours, If not, your battery may need replacing.
If you turn the key and the lights in the instrument panel dim but the engine doesn't start... or if the battery volt meter is in the red or low. This indicates that the battery is or may be dead and you need a jump. The only thing a jump can help you with is a dead battery. You may need a new battery if your engine has taken longer to start in the previous days/weeks. The battery may be fine and just need to be jumped if all you did was sayy... leave the lights on and run the battery down.
As long as the battery in the lawnmower is a 12 Volt battery you will be fine.
Does it crank over fine?
Battery
If you can jump start the car then the battery is not fine. Remove the battery cables and thoroughly clean them. If it still will not start check the voltage at the battery with the engine off. You should get 12.6 volts if the battery is fully charged. 12.4 it is 75%, 12.2 it is 50%, and 12.0 it is 25% charged. I would bet your battery is defective.
when you unhook the battery the computer probably resets itself.
Bad solenoid, bad ground, faulty ig. switch.
Have you removed the battery cables and cleaned the battery posts and cable connections? This is a frequent cause for the problem that you describe.
I would not recommend jump starting a motorcycle using a car, unless it's an extreme emergency. Then you do not want the car running, or you could fry your charging system. *Edit* Actually jump starting a motorcycle from a car battery is absolutely fine - whether the car is running or not. The motorcycle battery will only draw the power it needs from the car battery and won't overload it's own circuits. I have jumped a motorcyle from car batteries many times, and when we service motorcycles at the roadside that have dead batteries we hook them up to the generator in the van that we also use to start 18 wheelers that are having battery trouble, so its perfectly fine
If alternator and battery test fine- then you may have a bad fusable link not allowing the alternator to charge the battery