How long should I keep a cut onion?
When I use fresh onions, we store the unused parts for up to a week in our fridge in either a ziploc or a sealable rubbermaid-style container.  For particularly pungent onions they go in the crisper drawer to keep the smell from being too strong in the rest of the fridge, but usually they're just on one of the shelves.  Stilltasty says 2-3 days but my personal experience says otherwise.

Generally in our house, they don't last long enough for the flavor to degrade too much, but they will get a little dried and the smell will get stronger over time.  After a week, they shouldn't be bad in the sense that you'll get sick - but you'll want to use them in a meal where the onion is the star, as the aroma will take over the dish.

Another option that I'm a big fan of is to dice or strip the leftovers (or, a 5 lb bag at once!) and freeze them on a cookie sheet (instead of in a bag, where they'll turn into a giant clump).  Once fully frozen, dump them into a plastic freezer bag, and store in the freezer for easy access to a handful at a time.  For most applications, you can defrost them right in the pan/pot/casserole as you cook, so they're very handy and accessible.  We do the same with peppers in our house for convenience.