What are the differences between these types of small-dimension lumber?
There are a whole lot more types of boards than that! Even at the big box stores. 

Furring strips come in limited sizes and are not meant to be structural or seen.

Pressure treated boards have a chemical applied to them under pressure that helps presserve the wood against moisture and insects. It's generally used outdoors or in direct contact with concrete otherwise it's not recommended to use pressure treated lumber. The appearance, quality and sizes match other framing lumber used for studs, juists, etc. It would be cheaper to use non-pressure treated lumber.

The better wood to use are appearance grade boards. They'll either indicate they are "appearance" grade or described as S4S (surfaced 4 sides). These will have a smooth surface on all 4 sides. They come in a variety of different sizes and species. The cheapest price is for something the box stores call "whitewood" which is whatever light colored wood they can get the cheapest at that time. Pine, spruce, fir, etc. I think Home Depot might be calling them "common boards" now. 

The select grade boards are also appearance grade S4S boards but usually of better quality with fewer knots, hence the higher price. Not necessary for a garage cabinets.

That design you linked to looks like it would be expensive to build. If you want to save money and don't care about appearance you can save money by using a different design that uses 2x4's and 3/4" plywood for the shelves.