The alarm of my phone does not ring in silent mode. Why???

Because "silent mode" means silent - no sound. If you want the alarm to ring, get an app that splits notifications and alarms into the old 2 controls (they're a single control now - one of Google's "disimprovements") and turn off the other 3 - leave Alarms up. (Or use an app like Tasker so you have a "silent my way" button.)
 
Also check the phone settings: Sound-Quiet Mode- Block Alarms should not be checked.

Posted via the Android Central App
 
If you happen to have lollipop there is no more quiet mode... It uses lollipops built in dnd feature. Make sure you have "priority" checked and not "on ".

Posted via the Android Central App