Converting a time, not just an offset
A correct conversion needs three things: the wall clock time, the zone it was written in and the date. The date matters because offsets change. The tool takes the date and time you type, works out what instant that was in the source zone, then formats that same instant in the target zone. Under the hood it uses the browser time zone database through Intl.DateTimeFormat, which is why the result respects daylight saving rules, half-hour zones such as India at UTC+5:30 and 45-minute zones such as Nepal.
Worked example: 9:00 a.m. on July 15 in New York is Eastern Daylight Time, UTC-4. London on that date is British Summer Time, UTC+1, so the difference is five hours and the converted time is 2:00 p.m. Move the same 9:00 a.m. to January 15 and both zones are on standard time, UTC-5 and UTC+0, and the difference is still five hours. In early November, though, the US falls back a week after Europe, so for that week the gap is four hours. Converting with a fixed offset would get that week wrong.
US zones and daylight saving time
The continental United States spans four zones: Eastern (UTC-5 standard, UTC-4 daylight), Central (UTC-6 and UTC-5), Mountain (UTC-7 and UTC-6) and Pacific (UTC-8 and UTC-7). Alaska is one hour behind Pacific and Hawaii is two hours behind Pacific in winter and three in summer, because Hawaii does not observe daylight saving time. Most of Arizona does not either, so Phoenix matches Denver in winter and Los Angeles in summer. The tool lists Phoenix separately for exactly that reason.
Daylight saving time in the United States starts on the second Sunday in March and ends on the first Sunday in November. The European Union switches on the last Sundays of March and October, Australia runs its summer time from October to April, and much of Asia, Africa and South America does not change clocks at all. Those mismatched dates create a few weeks each year when the usual difference between two cities is off by an hour, which is the single most common cause of missed international meetings.
Practical tips
When scheduling across zones, write the time with its zone abbreviation and the UTC offset, for example 3:00 p.m. EDT (UTC-4), so no one has to guess. For recurring meetings, pin the time to one city rather than to an offset, since the offset will change twice a year. Store timestamps in UTC in any system you build and convert only for display. The city table on this page shows the same instant everywhere at once, which makes it easy to find an hour that is reasonable for everybody.