Foil kites are soft kites based on the design of the parafoil. They consist of a number of cells running fore to aft, some or all of which are open at the front to allow air to inflate the kite so it takes on an aerofoil section.
Open-cell foil kites have an open structure and can be used on land or snow. This is the most popular model if you want to kite only on snow. This is the easiest model to launch and land on your own. Very good option for backcountry snowkiting and kite expeditions. Open-cell foil kites are not suitable to be used on water.
Closed-cell foil kites have half closed structure. In order to make them suitable for use on water closed-cell foils have limited air inlets in the center of the leading edge, with valves to keep the air in and (hopefully) the water out. Internal holes in the cell sides allow the whole kite to inflate. These kites are naturally slower to inflate than an open-fronted foil. Closed-cell inflatables can be used on water, snow and land. Usually not recommended for beginners.