To be popular or well-liked (especially with the opposite sex); can also mean being doted on or welcomed (colloquial).
彼は見た目がいいから女子にモテる。
He's good-looking, so he's popular with girls.
職場では仕事ができて、みんなからモテる存在だ。
At work he's competent and is well-liked by everyone.
金遣いが荒いと、長くはモテないこともある。
If someone spends money recklessly, they may not stay popular for long.
最近は服を変えただけで急にモテるようになった。
Lately he suddenly became popular just by changing his clothes.