Follow the (+) cable from the alternator to the battery.
At some point along that path there will be a fusible link, a length of wire with a different type of insulation.
When they blow, they are supposed to look burnt and stretch out.
Many times when they get this old they do not blow in the prescribed method, and can be intermittent at times. It might even test properly, but the second it sees any current it opens up and decides to pass no current.
Recharge your battery via a battery charger asap unless you like replacing batteries. Never rely on the alternator to fully charge a depleted battery, they do a very poor job bringing it above 80% unless you are doing some interstate distance driving.