To beg is to ask for something in an earnest, pleading way. You might beg your roommates to be quiet if you're desperately trying to get some sleep.
You can beg your parents to let you go to a concert with your friends, and you can beg your sister for forgiveness after you inadvertently hurt her feelings. In either case, you are imploring, asking sincerely for something you want. Many dogs are skilled at this, and will beg by sitting beside you while you eat, looking adorable. Another way to beg is to ask for desperately needed charity — to solicit money or food from strangers.