Hard Cheeses are pressed to remove as much of the whey from the curds, to help ensure a long shelf life. These varieties are matured for a number of months.
Two popular examples being Cheddar and Parmesan

Semi Hard Cheeses are pressed in moulds and are exactly as described - the texture being in between hard and soft.
Common examples being Gouda, Gruyere and Pecorino

Soft Cheeses can come in a few formats, either with a washed rind, natural rind or bloomy rind
Washed Rind - immersed in brine i.e Stinking Bishop and Saint-Nectaire
Natural Rind - Mainly made from sheep or goats milk, i.e Sainte-Maure de Touraine
Bloomy Rind -a white rind casing, these cheeses are made with either sheeps, cow or goats milk

These are the first stage of a cheese in so far as the curds are not ripened, pressed or cut and therefore can be eaten just a few hours after being made, such as
Mozzarella and Ricotta

These can be semi hard or hard cheeses which have herbs, spices or fruit within them, such as
Wensleydale with Apricot, Chilli Cheddar or Double Gloucester with Chives

Veined Cheeses, that contain mould. They can range from mild to strong, the 3 popular varieties being
Stilton, Gorgonzola and Roquefort