You'd be enplaned on a flight subject to the availability of space at departure time. This was only after all passengers with reservations for the flight have been boarded. Passengers from a previous flight who were bumped also had priority. No stopovers were permitted on standby fares. 

But the days when students could hang out at the airport and fly standby for dirt cheap prices are long gone. The term "standby" is still used, but doesn't mean what it used to:

1. Most airlines no longer sell standby tickets at a special fare, but will sell you a standby ticket at regular fare. In other words, if they have space on the flight, the gate agent or ticket agent will sell you a ticket for the flight on the spot. All this means is you can buy a full fare ticket on the spur of the moment.
2. Nonrefundable, nonchangeable tickets can often be used for standby travel on the same or similar route (sometimes with a slight surcharge). This most often comes up if you missed your flight (e.g., you arrive at the gate after the plane has left, and catch another flight to the same destination later that day), but you may be able to use unused flight coupons in this manner. For example, if you bought a ticket but didn't go for some reason (a reason that didn't entitle you to a refund), the ticket may still be usable for standby travel on the same airline. (Your best bet with such a ticket, however, is to have your travel agent or airline credit it towards your next ticket. You'll probably have to pay a fee, but it's better than throwing away the ticket.) All this is often at the whim of the gate agent, and is based on the notion that a ticket is often designated as being good for travel on the airline for one year from the date of issue. Your luck will vary by airline and gate agent. 
3. If you're on a later flight but get to the airport early, check with the attendant at the gate. You may be able to get on the earlier flight is there's space available (but this may result in your getting no "snack" or losing your special meal). This works even for "non-changeable" tickets. 

Airlines stopped the practice of selling standby tickets for a variety of reasons: 

1. Even if a seat isn't taken, putting somebody in the seat still costs the airline some money (e.g., fuel). 
2. Standby travel played havoc with airline load management programs. Airlines were sometimes faced with having to unload fuel to change the weight distribution of the aircraft because of a sudden appearance of several standby passengers. 
3. Some people would "ensure" the availability of standby seats by making a large number of regular reservations, and then an hour before the flight release the block of seats, virtually ensuring that standby passengers will get aboard at cheap standby fares. This caused airlines to overbook flights, and eliminated their ability to schedule their flights accurately. Since the elimination of standby tickets, overbooking errors have dropped significantly. (Another result of this practice is the automatic cancellation of unpaid reservations after 24 hours.)
4. Airlines felt that they were losing regular business to standby travel.

Flying standby, however, may be making a comeback. In 1993, some airlines had special one-day fares for folks traveling on Christmas and New Years. These holidays are the nadir of the travel season, so by offering these fares, the airlines were able to increase their business on flights they had to fly anyway. (Yes, airlines still have to fly the routes, even with empty planes, because scheduling is so complex. For example, the plane in Atlanta might need to fly to Boston so that it can be used on the Boston-Washington route later in the next day.) Now these holiday fares weren't true standby fares, but the concept was similar. Since most flights these days fly at 85-95% of capacity, one of these days an airline is going to get the bright idea of reinstating standby fares in one variation or another. For example, some airlines now sell discount flight coupon booklets to senior citizens; the coupons may be used for standby travel. And of course, US Air Force personnel can fly standby on some US Air Force planes on a space available basis for dirt cheap prices, but with no amenities (and maybe no real seat either, but strapped to webbing on the wall).

Some foreign airlines still provide standby fares, sometimes limited to youth. 

If you are flying standby one way or another, here's some advice: 

• Make sure you get to the gate EARLY. If several people are flying standby, you want to make sure that your name is first on the list. Note that connecting passengers, bumped passengers, etc., get priority over local boarding standbys. 
• Days which are bad for getting bumped are usually good days for standby seats. Don't fly standby on the day before Thanksgiving or the Sunday after, you won't get a seat. On Thanksgiving day itself, you're likely to find a seat. 
• On really busy days it might pay to show up early for the *first* flight of the day, since standbys who don't make it can "roll over" to the next flight. 

Some airlines will only allow "same day" standby travel (i.e., you overslept and missed your flight, and will take a flight on the same route later that day). Others will not allow you to fly standby on a flight if all the seats in your fare class have sold out, even if there are empty seats available. If the agent at the check-in counter gives you trouble, go directly to the gate and see what happens. [Agents at the check-in counter are under more pressure to enforce the rules than gate agents, who have some discretionary power. If you can avoid dealing with the agents at the check-in counter, (e.g., no checked baggage), so much the better.] If you want to try flying standby on one airline using another airline's ticket, bypass the check-in agent and go directly to the gate. 

Most airline employees (and their spouses and dependents) can fly standby on that airline for cheap or free (as "representatives" of the airline). Crew who must reach their destination for work purposes have priority over all non-revenue passengers. Active employees have priority over retired employees and dependents of an employee. There is a pretty strict dress code -- suit and ties for men, and similar requirements for women -- so you can't fly wearing a jogging suit.