Certified GKH Gardening Expert
Here are a few good articles on air pruning pots and felt pots:
They are especially good for plants that will be moved from the container to a new location eventually, because the roots will be more ready to start growing in their new environment. Felt pots can also be good if you want your plants or garden equipment to be more portable, since they'll be less heavy than traditional containers.
A few details I would mention: These pots will allow the soil to dry out more quickly, so they might be better for plants that like dry conditions or that you will definitely water regularly, and less good for plants that like more water, or that will be in a dry climate or in hot, bright sun.
Also, if the pot material is dark in color, it can cause the soil to heat up and damage the roots, so the dark colors are better for use in shade.