In most trees, the roots spread out away from the tree. Sometimes, however, the root will start curving inward and wrap around the trunk or stem, called a girdling root. As it grows around the tree, it can start to choke it and cut off nutrients. If the problem is severe enough, death can occur.
One of the most common reasons that girdling roots form is because the tree or shrub is being grown in a small space like a container or planting strip. As the roots move outwards, they hit the sides and begin to curve.
When I was doing my internship at the nursery, we worked on repotting containers. We would make quick slashes along each side to break up any girdling roots. This helps to curb the problem before they become large. When you buy a tree or shrub in a container, gently take it out of the pot to look at the roots. If the outside is full of curving roots, try to find one without that problem, though it can be hard to avoid. You can also have a lesser chance of this problem by buying bare root or ball & burlap trees and shrubs instead.
When you are planting, use the flat side of some hand pruners (or even something like a box cutter) to make a few cuts down the side of the root ball and across the bottom. This will help stop any roots from girdling (though this is not foolproof) and it will also help new root growth to start, helping your new plant get settled in faster.
If you discover than an established tree has a girdling root, you can carefully prune it out. In some cases this can stress the tree or lead to its death, but the girdling root will eventually do that anyways. Pruning away the offending root can potentially save your tree.