Licensure by endorsement is done when you've already passed the NCLEX (RN or LPN, whichever applies to you) and have been licensed in whichever state you are leaving to practice elsewhere (say, State 1). Basically to get a license by endorsement, you have to already have a license in good standing with State 1. When you go to apply for a license in State 2, applying via endorsement allows the board to essentially check your credentials with the State 1, your school, police, etc., to make sure you are what you say you are (licensed to practice elsewhere). You pay a fee to State 2 for the license if you're approved (these fees are typically not refundable). State 2 will let you know if you're licensed, and you're all set to practice if you are approved.
If licensure by endorsement wasn't an option, we'd all have to rewrite boards....and that would suck.
To keep any license, you have to maintain whatever requirements the state BON attaches to having a license. For active licenses, sometimes this is just paying a renewal fee. You can be licensed in multiple states at one time. You just have to pay all the applicable fees and meet whatever other requirements each state's BON requires to maintain a license in that state. In New York, there is a CEU requirement and a fee. In New Hampshire, there is a CEU requirement, a fee, and a criminal background check. In Vermont, there is a fee only. I know because I had license in all three of those states (plus two others) up until last year. I still maintain licenses in 4 states.
If you reside in and are licensed to practice in compact state, then licensure by endorsement is a bit different. For example, New Hampshire and Maryland are compact states. If you are licensed and reside in New Hampshire, but plan to relocate to Maryland and would like a license by endorsement, you have generally a month where you can practice on your New Hampshire license in Maryland. Apply for a Maryland license by endorsement before the end of the 30 days.
If you reside in a non-compact (licensed to practice in a single state only) state and are desiring a license in another non-compact state, you cannot practice in the other state until you have a license for that state.
If you reside in a compact state and are desiring a license in a non-compact state, you also cannot practice in teh other state until you have a license for that state. Also, since your state of residence will change if you move to the new state, your license in the compact state is converted to a non-compact (single state) license.