In the olden days, food was pureed by pressing it by hand through a fine mesh strainer. This is still a good method, especially for foods that can be altered by excessive beating, like potatoes. But most cooks lack the patience for such antiquated methods (Wimps!) and prefer more electrified means.
Blenders make the finest puree. Because the container is cylindrical and narrow, the food is forced into the blade frequently. A food processor, which spins the food toward the outside of the container, gives slightly chunkier results.
Immersion blenders are basically blenders on a stick. A little propeller is inserted into a pot of food and as it whizzes around, the food is drawn into the blade. The advantage is that the soup does not have to be transferred out of the pot, and only the blade of the immersion blender needs to get cleaned. The downside is that is a little harder to get a really thin, fine puree.
Food mills are an old-fashioned manual option that work really well, but take a little extra elbow grease. Regardless of the puree method you choose, it’s a good idea to cool the food as much as possible once it is cooked before you puree it. The splatter potential is real, and the results can be scorching. Use caution. Many cooks forget that steam is hot and hurts just as much as boiling water, if not more.
Once your puree is finished, you may wish to strain it to remove any oversized chunks. This is how the smoothest possible purees are made. Use a fine mesh strainer, and press the mixture through with a ladle.