Learn French: Il ne met jamais son sac ici. - He never puts his bag here.

The verb mettre (to put) is irregular. In the present tense for singular pronouns like je, tu, and il or elle, the base loses one "t". For "he", the conjugation is il met. Notice there is no "s" at the end, unlike je mets or tu mets.

Why is there no pas in this sentence?

French uses two-part negations. Usually, you see ne ... pas (not). But when you use jamais (never), it replaces the pas. The ne and jamais hug the conjugated verb just like ne ... pas does. You would never say ne met pas jamais.

If we changed the sentence to "She never puts her bag here", would we use sa sac instead of son sac?

No, it would still be son sac. In French, possessives agree with the gender of the object being owned, not the owner. Since sac is a masculine noun, it is always son sac, whether it belongs to a man (his bag) or a woman (her bag). You would just change the first word to Elle.

Besides placing an object somewhere, what else is the verb mettre commonly used for?

Mettre is incredibly versatile! In addition to placing things, as with the bag in this sentence, it is the standard verb for putting on clothes. For example, Il met un pull means "He puts on a sweater."