There are five basic types of military discharges: honorable, general (under honorable conditions), other than honorable, bad conduct and dishonorable. Each of these denotes the quality of service and discharge conditions under which the service member completed his contract.
An honorable discharge is given to service members who performed above and beyond the call of duty and who were on excellent terms upon discharge. General (under honorable conditions) means the service member successfully completed his contract, but there were circumstances that were less than honorable either during his service or upon discharge. An example of this is a service member who is discharged prematurely due to a medical condition or other unavoidable situation. Other than honorable, bad conduct and dishonorable are all discharges which mean the service member didn't satisfactorily complete his service term. Going absent without leave or being found guilty in front of a court martial warrant these types of discharges. Bad conduct and dishonorable discharges are only given out on the order of a court martial or other equivalent proceeding.
You can easily identify a person's discharge status by checking the military DD-214 form. This form lists the discharge type, length of service, duty, and any mitigating circumstances of the discharge. Mitigating circumstances could include "Signed in Absentia," which is an indication that the service member was absent without leave or missing in action when discharged.
A service member who is discharged with an honorable discharge is usually allowed to reenlist if she is still eligible due to age, medical conditions and physical ability. In some cases, a person with a general discharge or other than honorable discharge can reenlist by obtaining a waiver. Bad conduct and dishonorable discharges prohibit reenlistment.
Honorable and general discharges are totally different. The wording "general under honorable conditions" can easily be misinterpreted to mean that the general discharge is equivalent to the honorable discharge. This is a big misconception. An honorable discharge means the service member has an excellent military record for the entire length of service. A general discharge means that the service member completed his obligation to the military, but that the service wasn't excellent. The one thing both of these discharges have in common is that service members with both of these discharges are eligible to receive all benefits due to them upon discharge, including the Montgomery G.I. Bill.
A service member's discharge can affect his ability to get certain civilian jobs. Some companies, especially security companies and companies that require background checks, may refuse to hire a veteran that was discharged under other than honorable, bad conduct or dishonorable conditions. In some cases, even a general discharge can sabotage a civilian job. Many civilian employers consider an honorable discharge a big plus.