Firstly, it's not a mini LP replica - it may look like the original release and may even in some cases be superior (or at least have more features) - but it's not a mini LP replica.

In terms of CD packaging, most CDs are sold in "jewel cases". The next most common packaging is probably digipack.

Digipak is actually a brand name. They were the first company to produce the packaging. It usually has a cardboard outer cover that always folds open.  Inside there is a plastic tray stuck to the inside of the back cover.

The biggest drawback with digipaks is that you cannot replace the CD tray. Once the little teeth / sprockets that hold the CD in place are damaged then the CD freely floats around (and invariably falls out).

Be wary of buying digipaks through the post.  If you sell them on ebay and want to avoid hassles after the sale, place some bubble wrap (one square inch is enough) inside the cover over the sprockets before packing.