"Please give me a bag."
Theoretically 包 is also ok for 'bag'. But this sentence is mostly used in the context when you check out in the supermarket and ask for a plastic bag, in this case it should be "袋子” , not 包。
包 is usually used for bags with closing such as 书包school bag，背包backpack，手提包handbag，钱包wallet. When you say "我的包呢？ (where is my bag?)”, people would assume you are talking about a backpack or handbag.
袋子 refers to bags without closing, such as 塑料袋plastic bag, 购物袋shopping bag, 纸袋paper bag.
One typical example would be 手提包 and 手提袋，put them in google image search and you would see the difference of the concept :)
包子is a food as Keith said.
Many Chinese words are made by a core character and some other attribute characters to form compound nouns. 包 itself would point to bags in general unless it is understood in the context e.g. 我老婆到巴黎買包去了 / My wife has gone to Paris for buying bags. In this sentence we won't think about bags other than luxury handbags. Otherwise, to be specific we need to use those compound nouns: 背包/rucksack, 手提包/handbag, 書包/school bag, 公事包/briefcase, etc.