Translation:I will give a watch to my mother.
Japanese does differentiate between ones own mother (はは) and other mothers (おかあさん).
Taking it a bit further:
-eru and -iru ending verbs you remove る to add -ます. Ex: 食べる (たべる) becomes 食べます (たべます).
-u ending verbs (including the other -ru endings) you change i to u before adding -ます. Ex: 上る (のぼる notice the -oru ending) becomes 上ります (のぼります).
And then there are the iregulars -uru endings する and くる, where you remove the whole -uru to します and きます and yeah, good luck.
"Offer" implies you are giving a gift or something special, or you are trying to make a deal with someone. It heavily implies that the recipient is being given a choice of whether or not to accept the item. "Give" is much more simple, and tends to describe the action itself without regard to the circumstances surrounding the action.