Trainee Pilot reported alternator failure resulted in a diversion and a precautionary landing.

Narrative

Planned route was ZZZ-ZZZ1 to complete a XC (Cross Country). Around [Location] we (me+CFI) got a Master Warning; I did not catch what the message was but volts were highlighted in red and were low (between 24 and 25). My CFI ran an abnormal procedures checklist while I continued to look for my point. I started to smell smoke and asked her if she smelled anything; but she didn't so I thought nothing of it. Maybe 30 seconds later she began to smell it and smoke came out from under the panel on the left side. She took controls and made a call to Center; who we were talking to for flight following. I ran the electrical fire checklist. She requested priority handling and we turned in the direction of ZZZ2 because it was about 6 nm south from us which was determined after the fact using her Foreflight track. After completing the checklist she had me open my window to aid with venting the cabin and shortly after the completion of the checklist the smoke went away. She descended; then overflew the field and entered left traffic for Runway XX. Below 2;700 ft. or so Center could no longer hear us and we switched to the CTAF for ZZZ2 before turning final where we communicated with Aircraft Y who relayed that we had landed to approach. We taxied off and tied the aircraft down as normal and closed our flight plan over the phone. So far from what I know a wire broke off the alternator which caused a fuse to break and that fuse was causing the smoke. Airplanes that do not break; which don't exist but the encouragement by Flight School to memorize checklists helped; as did the proximity to ZZZ2 airport.