What can I do about these tantrums?
Tantrums are my kryptonite. I hate tantrums! I have taken care of many, many children, and all of them have had tantrums of some sort. I choose not to respond to tantrums. Tantrums that aren't about hunger or sleepiness, are completely ignored. I try and only give positive attention, because I don't want to teach the kids that throwing tantrums is the way to get my attention. My 7 year old still has occasional tantrums. I ask him to go to his room until he is able to talk calmly with me, and discuss whatever the issue is appropriately. It usually only takes him a minute to calm down and be able to talk. But what I am trying to instill is that acting out is not the correct way to get my attention.
Be patient, but be consistent, but try hard not to respond. Give praise when praise is due, and ignore the rest.
If there's nothing wrong with him/her (not hungry, not in pain, etc.) and they're just angry/frustrated... ignore the tantrum. Put them in their crib or pack-n-play or whatever you use for 'time out' and simply wait for them to finish throwing a fit. Your kid *will* get tired of screaming eventually, the hard part is waiting it out without going nuts :/ my daughter is 2 and she's had some pretty intense screaming fits when she doesn't get what she wants, but as soon as she starts it she goes right to the pack-n-play. I tell her, "When you're ready to calm down and be nice, I will get you."
It seems harsh (and it breaks my heart to hear her cry), but if you give in you just teach them that they just have to scream long enough and they'll get what they want no matter what you say - which is NOT something you want to reinforce. My daughter's tantrums don't last anywhere near as long now as they used to because she's starting to understand that screaming and stomping around just gets her time-out.