Battery vs Alternator

Your alternator is most likely bad. The easiset way to find this out is to recharge your battery fully, get a voltage reading once that's done and it has sat for a few hours. The battery voltage reading should be something above 12,4 voltsor so-if its NOT, then your battery cannot hold much of a charge. If that's the case, get a new battery. IF the BATTERY voltage is above 12.4 volts and the car starts, keep the voltmeter hooked up and watch the voltage on the voltmeter while the engine is being started and then as it runs. The voltage while the engine is cranking over should momentarily drop and then shoot back up as soon as the engine starts. While the engine is RUNNING, the battery voltage should be higher than when it is not-and that voltage reading should NOT be dropping even if you turn on your headlights or the a/c. If it DOES drop while the engine is running continuously that means the charging system is not capable of keeping the battery charged under normal conditions. As to the jump start. Its quite possible that the jump vehicle's charging system put enough energy into your battery to run the ENGINE for much more than 5 minutes, as the engine itself does not draw very much energy from the battery. For future reference, the a/c may use up energy at 20x that rate or more, as may the headlights (if not LED's), the heater, or the window defroster, (OR a massive sound system). So if there is any doubt about your charging system being able to keep the battery sufficiently charged DO NOT USE THOSE ACCESSORIES until the situation is addressed

Test the alternator and battery. Your local auto parts store can do this for you at no cost.