I work with toddlers so I see quite a few tantrums. I have not read the other responses, but here is my advice:
At this age, your daughter has strong emotions and is not quite sure how to show or handle them. So, they erupt as tantrums. You can do several things to help her get through this time.
First, understand that she does not know how to effectively and appropriately handle all these strong emotions. This skill has to be taught. You can help her learn how to do this. It is a learning process. She may need some time to get the feelings out through a tantrum, just ignore it, let her do it and make sure she is safe. Definitely DO NOT give in. This teaches her that what she is dong effectively gets her what she wants.
After she is calm, calmly but firmly talk to her about her feelings. Say, "I understand that you are very sad/angry/mad. It is not ok to scream and behave that way. When you are angry/mad etc. you can use your words to tell me (or whatever you want the alternative behavior to be)." Basically, you are validating her feelings (remember, it is ok to feel that way, but there are appropriate and inappropriate ways of handling those feelings); telling her what is not acceptable; and then giving her an alternative and acceptable way of handling her feelings next time. Be consistent! Validating her feelings and using words to describe them helps her make sense of all the confusing emotions.
Before conflict arises, you can give her choices to avoid the unacceptable behavior. "Do you want to clean up your toys by yourself or do you want me to help you?" Either way she is cleaning up and it doesn't really matter if you help or not. Give her two choices that you are ok with and let her choose. Letting her have more choices throughout the day can help her feel a little more control over her world, but not an overwhelming amount. Other examples might be, "Would you like greenbeans or peas with our dinner tonight?" "Would you like to wear your blue pants or green shorts today?" Then, stick to your choices. If you see it may turn in to a tantrum then at least make her ask nicely (using her big girl voice). Don't let the fit be the last thing that made you fulfill her request.
I think one of the main keys here is to be consistent. You have to follow through with what you say. At her age, and any age really, a pat on the hand or time-out does not teach a child what they should be doing instead, only what they shouldn't do. Try to focus on what she CAN do, like "I can't let you have candy at the store, but we can have a snack in the car. Would you like a granola bar or fruit in the car?" Like I mentioned earlier, after she has thrown a tantrum and you ignore it, calmly but firmly tell her that she can use her words when she is angry to tell you how she feels. Give her examples of what she can say and practice saying them with her. When you see her start to get upset, you can calmly remind her what she can say. Then, figure out a solution that you can both agree on. This might sound like, "I understand you want to play outside right now. That does sound fun. Right now it's nap time and as soon as you wake up, we can get shoes on and go out. Would you like to even eat your snack outside too?"
Another thing that might help avoid some of these situations is telling her ahead of time what to expect. Like in the examples above, remind her BEFORE you get to the store that you are not buying candy. And, talk about the plan for the day in the morning, so she knows she can play outside after nap. Giving her an idea of what to expect ahead of time in most situations can help avoid tantrums.
Remember that having a strong will is not always a bad thing. Later in life, she will never be a pushover. Your job right now is to help her learn how to use that strong will in a useful and respectful way. As an adult, you would probably want her to be able to voice her strong opinion with co-workers respectfully without offending them, and also be able to get her ideas carried out. Think about what you can do NOW in order to hep her learn these skills. Help her to also learn about other's ideas and opinions as she grows. Look for opportunities where you can describe what another child may be feeling. For example, if at the playground you see another child get pushed or hurt, explain to her that the child must be feeling pretty sad right now. What's something you could do to help them feel better? Talking about and naming her emotions and other people's emotions (on TV, in public, etc) can help her work through this.
You can also model respectful and polite behavior in your tone of voice and they way you talk to her. This will be the best example for her.
Hope this helps a little, let me know if you want to talk more specifically about your daughter. I have helped many families with tantrums and it's great to see a positive result!
Best wishes! C.