A puree is a very smooth, crushed or blended food — like applesauce or mashed potatoes. To puree something is to blend, chop, mash, or strain a food until it reaches this soft consistency. If it tastes like mush, then it's a puree!
You might think of purees as baby food, and it's true that a vegetable or fruit puree is the perfect first food for a baby to eat before she has teeth. There are plenty of things that adults like to puree too, though, like chickpeas (to make hummus) and avocados (to make guacamole). Puree is from the French purée, "pea soup."