Sometimes aloe grow that way. There are many different varieties of aloe. You can remove the offshoots (pups) once they have their own root system and pot them up. You can't take a cutting just from a leaf as far as I know. You should find these helpful. Take a look at the first site to see if you can id which aloe you have. This is only a list of a few that are out there. Some of the pics have numbers below them with additional pics.
Click on 'A' at this site and scroll down to 'Aloe' and the different varieties.
If you still need help, let us know.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.