Daylight saving time (DST), or "summer time" in Europe, is the time of the year when certain parts of the world set their clocks forward one hour during the warm season, in an attempt to "save" more daylight for evening use during the long summer days, instead of wasting it on the early morning, when people would still be sleeping.
Much of our modern technology (computers, phones, game consoles, etc.) will automatically adjust for daylight saving time. For older or analog timekeeping devices that must be manually set, the handy mnemonic "spring forward, fall back" can help you to remember that the clock moves forward come springtime, and back in the fall/autumn.
In the U.S., since 2007, daylight saving time has, by act of Congress, begun on the second Sunday of March and ended on the first Sunday of November. (So, for 2013, DST begins March 10 and ends November 2.) Typically, the time change in each case is handled at 2:00 am. That is to say, in March, the clock will go from 1:59 am to 3:00 am, while, in November, it will go from 1:59 am to 1:00 am.
If you're having to set the clock on a device at any other time of year, and it's asking you whether DST should be "On" or "Off," just remember that daylight saving time was specifically conceived for the summer (hence its name, "summer time," in Europe), and so it should be "On" between March and November. Often, you'll be prompted to set this right after setting the date and time, and the default will be "Off," in which case switching it to "On" may instantly add an hour to whatever time you previously entered. This is because, when daylight saving time is "Off," we're officially operating by "standard time" instead (making daylight saving time technically the deviation from the standard). Finally, if you have one of those TVs or VCRs or whatever confusingly asking you whether "summer time" should be "In" or "Out," it's probably not made in America, so remember that "summer time in" is just a British English way of saying "daylight saving time on."