Today's prayer times, wherever you are
Fajr, Sunrise, Dhuhr, Asr, Maghrib and Isha for wherever you are, worked out from where the sun is today. It is all done here on this page, so your location is never sent anywhere.
The adhan on time, wherever you are
This page can show you the times, but it cannot call you to prayer: close the tab and it is gone. The free Android app can. The adhan sounds at the right minute for every prayer, with the app closed and your phone in your pocket, in whichever of the 28 recordings by 24 muezzins you choose: 17 for the daily prayers, and 11 made for Fajr. Sunrise, which is not a prayer, gets a struck bell of its own instead.
Your city, already worked out
If you would rather not hand over your location, or you want the whole month at once, there is a page for 340 cities with the timetable and the qibla direction already on it.
Questions
How are these prayer times calculated?
From where you are and where the sun is today, using the standard angles for the method you choose. It all happens on your own device, and your location never leaves it.
Which calculation method should I choose?
The one your local masjid uses, so your times match the jamaah. Muslim World League is the usual choice in Europe, Umm al-Qura in Saudi Arabia, Karachi across Pakistan and India, ISNA in North America and Egyptian in much of Africa. They differ mainly in how far below the horizon the sun has to be for Fajr and for Isha.
What is the difference between Hanafi and standard Asr?
Both watch the shadow of a standing object grow through the afternoon. In the majority position Asr begins once the shadow has grown by the object's own length, and in the Hanafi school once it has grown by twice that, so Hanafi Asr comes later. Follow whichever your masjid follows.
Do I need to allow location?
Only if you want the times for exactly where you are. You can type in a latitude and longitude instead. Either way, nothing leaves this page.