If you jump start, the battery will need engine running time to recharge ( like an hour + ) . Connecting the battery to a charger would be a better choice as this will lessen the load on the alternator.

If a battery is deeply discharged, it will be damaged. A newer battery can tolerate this but as the battery ages, it becomes more prone to permanent damage.

How a lead acid battery works in basic terms.

LA batteries are a reversible chemical reaction. Two porous lead plates seperated be a paper ( ish ) sheet are submerged in sulfuric acid. One plate is considered positive and the other negative. When an electrical load is placed across the plates, sulfur migrates in to the plates, this also reduces the acid rating of the liquid. When a battery is recharged, sulfur is forced out of the lead plates and back into the liquid.

When a battery sits in a discharged state too long, a crust of sulfur builds up preventing sulfur from being pushed out of the plates during charging or being absorbed during discharge. This is called sulfation.

In order to make 12 volts ( of electrical pressure ) , there need to be 6 cells with separate acid wells. In order to give enough amperage ( electrical volume ) there needs to be high surface ares of the plates. This is done by interleaving more positive and negative plates in a single cell / acid well.

A battery can fail to provide any voltage due to breaks in the internal connecting structure. It can have reduced voltage if a cells plates touch due to a failed separator. It can have reduced available amperage because lead has fallen off the plates or the are blocked by sulfation.

A load test measures available amps ( how big the gas tank is ) , a voltage test measures state of charge ( how full the gas tank is. )

If a battery has not been charged or discharged in 8 hours +, this chart will tell you how full the battery is. ( Checking voltage while the engine is running isn't a valid test as charging voltage needs to be higher than static voltage to push sulfur out of the lead 13.7 to 14.5 ish is typical .)

12.66 100 % charged
12.42 75% charged
12.18 50% charged
11.94 25 % charged
11.70 0 % charged ( going lower will damage the battery )