Natural born hunters
When your kitten jumps and attacks the heels of a visitor, it’s exactly how she would attack prey. While she's young it’s usually just a game, and her nips and bites are little more than playfulness. As she grows up though, those little bites can start to get quite forceful, so it’s always a good idea to interrupt her game and ignore her for a while. If this doesn’t work, speak to her with a firm, loud voice (but don't shout!). She’ll soon learn the association.
Sometimes your kitten will show a different kind of aggression, when she sees another cat but can’t reach or attack it thanks to those wretched modern inventions like windows and gates. The best thing to do in this situation is to leave her alone to calm down. By the time something more interesting grabs her attention she'll have forgotten all about that elusive cat.
She may be your adorable kitten, but her natural hunting instinct can still be quite upsetting, especially when she brings her dead prey home. Punishing her for doing it, of course, is wrong - she's just following a deeply rooted instinct, using skills she learnt when she was a tiny kitten. It’s certainly nothing to do with being hungry. She'll carry on hunting however well fed she is, and may even present her prey to you expecting to be congratulated.
There are, however, steps you can take to discourage her hunting expeditions. Play with her enough and you might become far more exciting than the birds in the garden at night. Another good, traditional idea is to put a bell on her collar, which will warn birds and other prey that she’s out and about on the prowl.