Now the multi-piece jigsaw puzzles I have always stored in little zip-top baggies with the image cut of the box or a picture of the complete puzzle in the baggie. I also like to color-code the puzzles which is actually pretty simple. Just put the puzzle together and then carefully flip it over and color the back with a permanent marker or even a washable one if the back is rough cardboard (if it's a laminated or white shiny back, use a permanent marker, you will thank me later). This way, if several kids are using puzzles all together and in roughly the same space, they can easily figure out which pieces go with which baggie. You can also add that color to the baggie too to make it easy.
The thing that has kinda always frustrated me is our board puzzles. I have a couple of those awesome metal puzzle racks because I love utilizing vertical storage to maximizing our limited space in our classroom. I purchased one and was gifted another specifically to help aid in the organization of those darn board puzzles however my students just can never get the hang of putting them away correctly.
So this is pretty much what our board puzzles look like most of the time. They are either put in the rack crooked or just laid at the bottom length-wise and stacked on top of each other. UGH!