Some imports have a adjuster that is under the pivot bolt...check this.Also there may be a nut on the back side that is spinning. The threads are not left hand.
There is a belt tightener on the alternator which must be loosened to take the belt off. After the belt is loose, remove the bottom mounting bolt and the alternator is off. you will also need to remove the wiring.
Not knowing the year, make, model and engine I suspect you have more parts to remove.
Alternator belt is probably loose and slipping. If you can turn the alternator by hand and the belt slips its too loose. Retension alternator belt
Corroded or loose battery connections, loose alternator drive belt, defective alternator, or dead cell in the battery.
alternator, or a wire to alternator loose
Can be loose or corroded battery connections, loose drive belt, or the alternator is failing.
You will need to loosen the alternator belt with the tensioner pulley. When the tensioner pulley is loose the alternator belt will become loose. Put the new alternator belt on and tighten the tensioner pulley.
I assume it is V-belt drive, is the top of the alternator sliding in the slotted bracket? (check if you need a washer or two, or stripped threads) but sounds more like a cheap belt that is stretching.
loose rocker arm[s]
Make sure both the top and bottom bolts are tight, if the bottom is loose the alternator can "wiggle". Also I have noticed that the top bolt may sink into the slot provided for alternator belt adjustment. I put a slightly larger diameter thick washer on the top bolt and it cured "the agony of the screech"!
Have the charging system checked. Possibly loose or corroded wiring, a loose alternator belt, or a bad alternator.
By looking for any loose threads