How thick can it cut?

Diode lasers are predominantly designed for engraving. Their cutting capacity is limited. Cutting repeatedly will greatly shorten the life of the laser due to thermal stress on the circuitry. 
Generally, the laser diode can cut thin woods and will require multiple passes to do so. It can cut up to 3mm balsa, 2mm cardboard, 4mm black foam, 1mm mdf and 3mm paper.
User experience will vary depending upon laser power and speed, as well as materials.