Tantrums are SO normal. Uhura is right that it is in part b/c they don't have words (some do, though, like my oldest who at 14 months could say, I'm mad at you mama). And I don't think it's ever something to be embarrassed about.
Having been a nanny for many years and now a mommy, I have tried and true tactics for this. One is to say, calmly, "you keep doing that, honey, and come find me when you're done." Then walk away. You'd be surprised at how fast the kid regains control when he/she doesn't have an audience!
Obviously you can't walk away from a dangerous situation. In that case, and this is number two (duh, Mandy), is to remove the child from the danger spot, put them somewhere safe and repeat number one. I say something like "we don't have fits on the stairs/in the bathroom/on the sidewalk." It works and you'll find that you don't have to do it too often before they realize a fit is not going to work in their favor! I've even said, "you can't have a fit here but as soon as we get home I'll let you have a big one." By the time we've gotten home she's either forgotten or has informed me that she doesn't need to have one anymore.
My oldest is now 3 and very precocious. When she was 18 months (and I remember b/c I was hugely preggers with her little sis) she saw a little boy having a full-blown tantrum in front of the milk at the grocery store. The poor mom was in tears and M said in her most authoritative voice, "He's having a conniption. He should have his conniption at home instead so he's safe." Well I was mortified but the mom started laughing so hard. She told M she was completely right then looked at her son and told him she was leaving and he could come find her when he was ready. OMG, his reaction was priceless and she learned how to deal with the fit. It still embarrasses me a little though
Even a 12 month old can understand no and how to manipulate to get their way. Being firm with our children is one of the only ways we have to deal with them that lets them know that we are in charge. Remember that they need for someone to be in charge and, while it makes them upset at times, they look for us for that safety. Oh, but if a fit has escalated to the point where the child just can't stop crying, and that happens too, when they are crying and no longer know why they are doing so, then hold that baby!