In my country, the official time changes in the first month of spring and fall. When the official time changes, my mobile phone's and PC's clocks change as well.
How do these devices find out there's a change?
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Most operating systems have the basic rules to calculate this (e.g. first weekend after a specific date), so they can adjust on their own, without requiring a connection to any time server. This can however screw up the system's clock in case there are changes of the official rules (e.g. the latest changes for Russia) or a system backup is restored (so the system adjusts the time twice).
As an alternative, some devices might contact a remote server (they most likely do it all the time anyway) or use some other kind of embedded timestamp (e.g. broadcast as part of video text/ceefax or DVB data, when talking about television programs and receivers/recorders/TVs).
Edit: Just to mention it: Older versions of Windows (e.g. Windows 95 or Windows 98) never trusted their own estimate, so they always told the user to verify the correct time once it changed. Windows Vista and Windows 7 (not sure about the versions in between) no longer do this and just assume they're right (in addition to using a time server to sync the clock).
Linux uses IANA time zone database to know when DST changes. A different story is for Windows for a number of reasons. One, Windows uses RTC in local time and Linux - in UTC. Imagine that a DST changes on Windows. This allows files to be "created" in the future and it may cause some problems. Also, older Windows treat DST differently, because then there were different DST rules and Microsoft does not provide updates for those Windows versions anymore, so they cannot now of changes in DST rules.
More on that here: http://en.wikipedia.org/wiki/Daylight_saving_time#Computing