Just to give everyone that might not know the full 411 on how Xbox Marketplace authorization works (at least as I understand it)... Your console has the licenses for your account saved and ANY account on that Xbox can use the purchases. However, your licenses can only be on one Xbox at at time (unless you keep the old Xbox offline, but as soon as ANY account is logged into Xbox Live, it will deauthorize it), and you can only migrate your licenses once a year. So, if you play your account from another Xbox where your licenses are not stored, then you can only play your purchases while logged into Xbox Live, and other accounts on the unlicensed Xbox cannot play your purchases unless you are able to have more than one account logged in (like fighting games that let each player have their own accounts)
This is correct but could be a little bit confusing for some so I'm just going to simplify it a bit for both the Xbox One and 360 as they have slight differences.
Xbox 360
All licenses for digital content are linked to your gamertag. This means as long as you purchased the content you will be able to play it on your account across any console.
The console that you purchase the content on also has a licence for that content. This means that anyone else's account can also use that content even if they don't own it themselves.
What you can't do is download content on another console through your account and let someone else play it while you are not signed in.
As mentioned by AbsolutGrndZer0, there is a way to transfer console licences from one console to another but there are limitations on this.
Xbox One
The Xbox One is very close to the same system.
All content you purchased is linked to your gamertag and can be played on any console as long as you are signed in.
The difference is that you can now set a "Home" console. This is a easy way to transfer all licences from one console to another and can be set from the Xbox One. Unlike the 360 which needs to be done through Xbox.com.
The "Home" console will also share gold with anyone else who is logged into that console.
The amount of times you can change your home console is limited as well.
A bonus of the Xbox One system is that if you have someone you want to share stuff with you can set your "Home" on their console and they can set theirs on yours. This means you both get full access to each others purchased content.
I currently share gold to my younger brother by using this method.