Choosing aircraft seat
Aircraft seats can be designated at booking, or at check-in at the airport. If seats are not designated, passengers take them by a first-come rule.
A few airlines do not assign seats (e.g., Southwest), but do assign you a boarding group based on how early you confirm your flight on-line within 24 hours of the flight.
As they receive your booking for a specific flight, most airlines will promptly assign you a seat. If so, visit its web-site soon after, and if dissatisfied with the automatic selection, see if you can choose another inter-actively from all available seats on the plane in your cabin class. If you are checking in at the airline's counter with no seat yet assigned, you should ask if a desirable seat is still available.
 Seats on a row
What's desirable? Different seat types on a plane have advantages over others:
Window seats are popular with many flyers, as they allow you to look out the window, rest your head against it while sleeping and not be disturbed by other passengers. The major downside is that you'll have to clamber over a seatmate or two to go to the bathroom or access any of your carry-ons in the overhead (although most airlines also allow you to stow baggage under the seat in front of you). You may also have slightly less floor space due to the curvature of the plane, and the wall can become quite cool.
Aisle seats are the choice of some road warriors, because they make it easy to get out and off the plane. On long flights, though, it's hard to sleep with people walking by, seatmates climbing over you, and the risk of an elbow being hit by a service trolley. Planes usually disembark row-by-row, so a seat further forward will often get you out at your destination quicker than an aisle seat farther back.
The third possibility is middle seats, which combine the disadvantages of both aisle and window seats without the advantages of either, although taller passengers may still be able to see the view from the window.
On well-equipped aircraft, some seats in each row may have entertainment electronics installed underneath. This can significantly compromise foot room for those seated behind. Poor foot room can be a major annoyance and source of discomfort on long flights. SeatGuru (noted just below) can help you avoid this.
There are some special rows too:
The advantages and disadvantages of various seats on many aircraft can be viewed at SeatGuru -- . Those wishing to avoid airsickness should choose seats over the wings of the aircraft, near the center of gravity. Occupants of those seats tend to feel less turbulence than passengers on window seats.
Passengers who want a bit more elbow room (even in economy class) should choose the last row of window seats on the plane. Due to the curvature of the plane, window seat passengers near the end of the plane may have a tad more elbow room on the window side...sometimes enough to fit a medium-sized totebag. However: