They're two different types of fried dough, sort of similar to doughnuts or funnel cakes.
Churros are Mexican and come in the shape of long hollow sticks (or in Spain, they're short flat sticks, but Disney has the Mexican kind) covered in cinnamon and granulated sugar; you dip them in bitter chocolate.
Beignets are from Louisiana and are shaped like pillows (they're hollow inside) and covered in powdered sugar.