Here is the fastest way I know of stopping a toddler's tantrum. Get down to their eye level, tell them they need to talk like a big boy/girl so you can understand them, then repeat what they say to you - stating it in a question form.
Here is an example of little Johnny wanting a treat at the store:
Mom: Johnny, you need to stop screaming and talk like a big kid so I can understand you.
Johnny still screaming...
Mom: Johnny, I can't understand you when you scream like that. You need to stop crying so I can understand you. (Keep repeating that until they calm down. Sometimes helping them with deep breathing will help too.)
Johnny, starts to calm down and says: I want candy.
Mom: You want that candy?
M: That candy looks really yummy doesn't it? (Johnny nods). We can't get candy today though. It is almost lunch time and we need to eat healthy things.
J: Starts up again.
M: I know you really want that candy, but we can't get it. Here do you want to play with this toy (or eat this other snack, or sing a song, etc.)? (Direct their attention to something else.)
We all want to be listened to and understood. Children do too. They just don't have the vocabularies or control to do it nicely so they scream until they feel like they are heard. Now it doesn't mean that they can/should get whatever they want, but they do need to feel like we at least understand.
Now, what if they don't stop? This is where you give them options. Let's go back to the story....
Johnny starts screaming again....
Mom: Johnny, I know you really want that candy, but screaming isn't appropriate.
Mom: Johnny, you will want to stop crying because you are going to want to hear the choices I am going to give you. You can calm down and eat this yummy snack I brought, or if you keep screaming I will take you home right now, and put you straight into your bed for a nap and you won't get stories. What do you want?
Johnny: I want candy.
Mom: I know you do, but that is not one of your choices. (Repeat choices)
Johnny refuses to choose.
Mom: If you don't choose, I will choose for you. If you keep screaming we will go home right now and you will go striaght to bed.
With the choices, I always present what I want them to choose and spin it to look as good as possible for them. Then with the other choice, I make it so that it won't be fun at all. They usually choose the first option.
However, they will push this to see if you will follow through so make sure you do. It will only take a few times for them to learn that you mean business and tantrums will be so much easier to deal with. If you don't follow through, they will learn that you are weak and they will start to push you around and they will scream louder until you cave in.