This type of rigs is used in shallow waters. They are submerged in the water so they are laid down on the bottom of the sea. That way the rig is moored although it may use anchors too.
Semi submersible rig is towed to location of drilling and then moored by combination of anchors and pontoons. It has a pontoons which are submerged to predetermined depth (usually about fifty feet).
It has a long legs which pass through the construction and lowers to the bottom. After jacking the legs down the platform rises above the water. When moving on new location the legs are jacked up and the rig is towed to location. Jack-up rigs can be used in waters to 500 feet of depth.
As the name says it is a ship equipped with drilling equipment. It is self propelled and can drill in very deep water. The ship is moored by standard anchoring system or dynamically positioned.
Platforms are structures made of concrete and steel and placed on the sea bed on site of drilling.