I can't speak for all the branches but in the Army there are five different ways to be discharged.
An honorable discharge means you did your service and your contract is up, this is the way you want to get out.
A general discharge can mean many different things from failing to adapt to the military, alcohol/drug abuse problems, or even medical issues. A general discharge in itself isn't necessarily bad, but depending on the reasons for receiving one you may end up forfeiting some benefits such as the GI bill.
3.Next there is a bad conduct discharge which is usually earned by breaking some military (but not necessarily civilian) laws. This type of discharge ends with some time in a military prison and loss of all veteran benefits.
Then we have the other than honorable discharge which is typically given to those that are facing legal trouble for civilian crimes and are facing jail time. They too will lose all veteran benefits.
Finally we come to the dishonorable discharge which you would really have to fuck up in order to receive. This discharge is reserved for those that have committed terrible acts such as murder, or desertion (different than AWOL). Once released from military prison those that have received a dishonorable discharge will have a rough go of things forfeiting not only their veteran benefits but many other freedoms as well. Similar to a convicted felon in the civilian world a dishonorable discharge recipient will not be allowed to legally purchase a firearm, will have a very difficult time finding work, and may even lose the right to vote.
I hoped this help shine some light on the subject for you.