Is phentermine a kind of narcotic?

No. Narcotics refer to opiate derived pain relievers. Phentermine is not one of these. It's usually used as a diet aid.
Maybe. It is not a narcotic by one definiiton that is a drug that in moderate doses dulls the senses, relieves pain, and induces profound sleep but in excessive doses causes stupor, coma, or can cause convulsions. But, a second definition is a drug that is subject to restriction like a narcotic. It is a controlled substance and that means it is restricted, in some ways.