I recently made an offer on my first home, and I was told by my real estate agent that the alley behind the house was part of the property, but the city took it to use it to read meters. Therefore, there is a lien on the property that will always be there, but it's OK , because it's really my property, but the city needs it.
What does this mean? I always thought a lien meant I owed somebody money and once it's paid off the lien goes away.

This is a different kind of lien. The city wants you to take care of that property as if it is your own but the lien prohibits you barring the city access. If they come to the point that they don't have to back in the alley to read the meters(like they have recently done at my own home) then they release the lien as they have no further need for access.

Is this a lien or an easement. It sounds exactly like an easement. The property is yours, but the city cannot be barred from using the property for their stated purpose.

As Scott said, it's not a lien, it's an easement. That means that you own the property but the city has the right to use it. You have to keep the alley clear enough for them to use it and you can't prevent them from using it.