In professional kitchens, remouillage or "remy" is a second run stock. Remouillage is mostly commonly used to start another stock, in place of water, or to make demi glace.
Remouillage is made by preserving the solid material after a stock is made and stained. All of the bones, mirepoix, and aromatics are returned to the stock pot, new cold water is added. The second stock is allowed to run for the full time of the original stock yielding a flavorful liquid fortified with collagen that that can then be strained and then cooled or frozen and used for something at a later date.
if starting a new stock from remy, load the bones and mirepoix into the stock pot, browning them first if making brown stock, and blanching the bones first of making veal stock. Next add the remouillage and fill the rest of the way with plain cold water.
If making demi-glace, the remy is used in place of stock, fortified with bones to help boost collagen and other aromatics, and then reduced down to the desired consistency. If making "old world demi-glace" you would use a blend of remouillage and sauce espsgnole.
Remy is not strictly limited to those classic uses. Bear in mind that it is a flavorful liquid - it can add background nuance to any braise, stew, or soup. Are you making chicken thighs in a recipe that calls for water? Perhaps a second run stock would produce a more flavorful braise. The cake can be said of veal, or beef, or pork.
The shelf life of remy is probably about 7 days under refrigeration, but under close inspection you can begin to see a degradation in quality around day 3. If you're not doing a regular schedule of stock production runs, then it's probably best to freeze it.