A flight from one city to another, without any stops.
Notable non-stops include the current world's longest flight from Singapore to Newark (route pictured above); the popular transcontinental non-stop of JFK to LAX; and many fun-in-the-sun long-hauls like London to Barbados.
Non-stop example (LAN from New York to Santiago)
A flight from one city to another, but which includes stops in one or more cities along the way. The flight number or aircraft remains the same, however, until the final destination is reached.
Notable direct flights include BA 001 from London's City Airport to JFK, with a refueling and immigration stop in Shannon, Ireland; South American domestic services like Cusco to Lima, with a stop in Arequipa; and the impressive JFK to Singapore SQ 25, with a stop in Frankfurt.
Direct example (LAN from Los Angeles to Santiago via Lima)
Ahh, let's not forget about connecting flights, which is likely how you'll be flying home for the holidays. A connecting flight is from one city to another, with a layover stop in between to change planes. Each flight requires a separate boarding pass, but they're on one itinerary.
Notable connecting flights include Virgin America's seasonal service from Palm Springs to any of their other cities, with a layover and plane change in San Francisco. Internationally, we did a doozy earlier this year by flying on Alitalia from JFK to Madrid, with a 30-minute layover and plane change in Rome.
Connecting example (Virgin America from Palm Springs to San Francisco to New York)
[Images: Jaunted, and maps created with Great Circle Mapper]