My DD4 has always been a screamer (literally came out of the womb screaming instead of crying like most babies... really she could be heard down the corridor). She had a few issues (cholic, terrible diaper rashes due to yeast infections and eczema, etc...) that only made her screaming worse... she learned that she got attention by screaming.
What I've done that seems to work... In the past year, as DD has gotten older, screaming, especially for minor things had gotten to be pretty bad. If DD was screaming for good cause (a big injury, etc..), then I would give her attention and encourage her to calm down. If DD was screaming for a minor reason (her brother took a toy, she wanted a tissue, etc..), then I'd tell her that she wasn't supposed to be screaming for no reason and would put her in her room until she was quieted down.
Now that DD is 4 she screams a lot less for minor reasons. But when she does, I tell her, "I can't understand you. You'll need to calm down and use your words to tell me what you want."
I think it's a little easier to reason with a child about this stuff when they're a little older. But even at 1 or 2, you can help yourself by not caving because of the screaming... if your DD screams for a toy or candy, tell her, "You aren't getting anything because you're screaming." Also, don't forget to give her positive reinforcement when she's behaving well... "You've been such a good girl today!"
If you give in too much to the tantrums, you'll only be setting yourself up for worse behavior later in life. Be firm but gentle and tell your DD when her behavior is inappropriate as well as reward good behavior.
Good Luck!