The thing a limit order buys you is a price, and the thing it costs you is certainty. Those are the only two variables, and every order type is a choice between them. A market order trades certainty of execution for whatever price is available at the moment it reaches the market. A limit order does the opposite. The choice therefore depends on which risk is the one that actually hurts in the situation at hand, and that varies more than people expect. Buying a widely traded fund with money that has to be invested this week, the risk of not trading is the larger one. Buying a thinly traded security whose quoted price moves several percent between trades, the risk of an unexpected price is larger.
A limit order is a standing instruction, and its duration is a separate decision that is being made whether or not you make it. The SEC states the default: "Unless an investor specifies a time frame for the expiration of an order, orders to buy and sell a stock are 'Day' orders, meaning they are good only during that trading day," and a day order that does not execute expires rather than carrying into after-hours or the next session. So an unfilled limit order does not quietly wait for weeks unless it was entered to. Brokerage platforms offer longer durations and other qualifications, and each of those is a further instruction with its own consequences.
Partial fills are ordinary and are the case people fail to plan for. An order for 500 shares can be met by a counterparty offering 200 at an acceptable price, leaving 300 outstanding. The result is a position half the intended size, and, on a platform charging per trade, potentially more than one charge. The general point is that a limit order describes conditions rather than an outcome, so the outcome can be none of it, some of it, or all of it.
A limit order is not a stop-loss order, and treating them as variations of one another is the most consequential mix-up in this area. A stop order is a different instrument with its own trigger mechanics and its own behavior once triggered, covered on its own page. The distinction that matters here is that a limit order is visible as a willingness to trade at a price you have chosen, while a stop order is an instruction that does nothing until a condition occurs. Assuming a stop behaves like a limit, or the reverse, is how an investor ends up believing they are protected in a way they are not.
Where the order goes after you enter it is a further subject. Retail orders are routed by the broker to a venue, and the arrangements governing that routing, along with the mechanics of how resting orders are displayed and matched, sit with their own terms. None of it changes the rule above, which is the part a reader needs before pressing anything.