Rendered Butter is a term with both an older meaning, and a more modern one.
Previously, rendering butter (melting it to remove the water and milk solids, e.g. clarifying it) was a good method of storing butter without refrigeration, because it would last longer. The chief goal was storage.
Now, Rendered Butter is more a generic term that includes concentrated butters, beurre cuisinier, or something between clarified butter and ghee, though most uses involve describing a product with a butterfat content over 85%. In Germany, Rendered Butter tends to be the same as concentrated butter. Rendered Butters can be stored or sold in resolidified form. In France, the term "beurre fondu" is used.
Literature & Lore
Procure as much country or Western butter as desired, you may get several pounds of it when it is cheap during the summer; or any butter unfit for table use may be made sweet and good for cooking purposes and will last for months, if prepared in the following manner: Place the butter in a deep, iron kettle, filling only half full to prevent boiling over. Set it on the fire where it will simmer slowly for several hours. Watch carefully that it does not boil over. Do not stir it, but from time to time skim it. When perfectly clear, and all the salt and sediment has settled at the bottom, the butter is done. Set aside a few minutes, then strain into stone jars through a fine sieve, and when cold tie up tightly with paper and cloth. Keep in a cool, dry place."
-- Florence Kreisler Greenbaum. The International Jewish Cook Book. SECOND EDITION. New York: Bloch Pub. Co., 1919.