Chorizo always refers to a pork sausage. The most common type is Spanish chorizo, a firm, red-colored pork sausage. Usually spiced with paprika and garlic, Spanish chorizo is cured and therefore doesn't need to be cooked (though it often still is). Sometimes, it's a little on the spicy side, but sometimes it's not. Like most types of sausages, there are as many variations of Spanish chorizo as there are people who eat it.

Mexican chorizo differs from Spanish chorizo in that it's an encased fresh sausage that must be cooked before eating. It also tends to be spicier (due to the addition of chile peppers) and fattier than its European counterpart.

Recipe: Chorizo And Eggs (Simply Recipes)
Reference: A Common Sense Guide to Chorizo and Spanish Sausages (La Tienda)
Reference: What is andouille? (Cookthink)