 and we are merely players This is not about me! I am not presenting my own work, receiving criticism for it and standing up in front of a scientific crowd. This is just a stage play. And I love acting. Today my role is that of a bright young scientist, presenting her work at a conference. The audience is going to be wowed by the performance. Whenever I get nervous and struggle my mantra is "This is just a play. There will be applause when it is over. Nothing here can hurt you because you are doing it for your own entertainment." It works. The earlier I can get "in role" beforehand, the better. And after the applause, you are back to your own self, as always.
The imaginary friend There are multiple options for this. If there is a person who's presence makes you calm, imagine them sitting in a place where you can see them. Smiling, thumbs up, positive everything is going to be fine. Or select an unknown audience member to take their role. Smile at them when you get nervous, imagine getting a smile back. It can make you feel more secure and the audience will receive you as very present and pleasing.
The power position The students are not listening or seem to be silently laughing at my teaching? I will show them who the lecturer is. This is my lecture, my topic and whoever is diminishing the wisdom I teach will have the results served to them at the exams and in the assignments. I am not that bad. I am nice to students. But sometimes being the hard one internally makes it easier to be the nice one externally. Because you know what you could do... although you never will misuse your power, of course.
With regard to my own experiences of music performance, I would say there are two parts to this. Firstly there is how well you know your material, as has been discussed. However, as you and I both know, no matter how well one knows one's material, one can still be anxious. This is where you can start to look after your physical and mental wellbeing through other means.
You may think it's unusual, but never underestimate the importance of diet on improving one's mood. Before my big, majorly important master's performance which I had been dreading for months, I looked after my diet a lot, especially the few days leading up to it. I didn't eat unhealthily, I ate a lot of oily fish [mainly salmon], which gave me a feeling of contentment and mental wellbeing. I also drank less. Additionally, physical exercise. I worked out the morning of my performance: this helped to take a lot of the physical tension out of me and to relax me more.
Additionally, deal with your emotions. If you feel upset, worried etc, don't be afraid to cry about it to let it out. About three weeks before my performance I had to do so [unusual for me], which was a lot of the tension coming out. But it was good for me and I did feel better afterwards. All of these things combined worked because I ended up getting a distinction in my performance.
So we can have therapy etc, but sometimes the simplest things - food, exercise etc - we discount because we forget how effective they are and we don't think they will make enough difference. We tend to think other things will be more effective, but looking after how we treat our bodies can have a massive effect on our minds.
Whether it may be mathematics, mastering an instrument or public speaking, to become proficient at any feat, one has to practice.
While there have been some clever suggestions made in the previous answers, what is surely to work is to continually expose yourself to this level of pressure and anxiety, either to the same degree, or gradually bit by bit, such as starting off with a presentation in a small group.
As an example from sport, the famous Williams sisters in tennis were subjected to verbal harassment by kids while they'd practice, who were paid by their father, to prepare them for playing in front of an audience. The point is not to go to such extremes, but rather that you can't expect to perform well from the get-go unless you learn to adapt these conditions.
Professor Cuddy from Harvard University, who has authored many papers and presented a TED talk on preparing for situations like public speaking and interviews, has many suggestions on dealing with these, such as what she calls power posing which noticeably decreases cortisol, a hormone related to stress.
I'll end with something tennis player Billie Jean King said, which is that pressure is a privilege. While these public speaking opportunities may be mortifying, that you are called upon to present your research and have an opportunity to contribute and interact with other researchers is a privilege, one worth learning to adapt to and perform well in.
I had similar problems with performing at folk clubs. Even though I knew all the people there, and I knew they were all my friends, it was still incredibly stressful to stand up and perform. This inevitably took its toll on my singing and playing, of course.
I found this self-hypnosis CD made a massive difference for me. In particular, there was one visualisation exercise which really took care of my stagefright. I still get nervous in public, but now it's a normal level of adrenaline which is simply excitement. I'm similarly more confident with public speaking as well.
I'm sure there are plenty of similar products around, and there may be some better tailored to public speaking. All I can say is that that's the one I used.
I'd start with professional help. Anxiety is often a treatable disorder, with cognitive behavioral therapy showing a success rate of about 50%. Start with a cognitive behavioral therapist. They treat anxiety disorders for a living. See where it goes from there.
Glossophobia (fear of public speaking) is the #1 fear in America. What has worked for many people (including myself) is to practice in a low-stakes environment until you build confidence. That confidence you want is sure to come the more you practice and refine your techniques. With enough practice, the nervous fear you experience now will turn into positive energy which makes your presentation exciting. The more nervous fear you experience now, the more energy you will have as a polished speaker.
A group like your local Toastmasters club is a great place to practice. If you do a search, you'll probably find a club very near you, possibly at your school.
Instead of using different psychological methods and drugs, I want to recommend you going for sports. And by that I actually mean any physical activities - even simple regular running will give you much more confidence. But to have better results, buy a subscription to a gym or start attending any sport training.
I am not talking about having some complexes about your appearance or anything, it's just about how you feel your own body, because that's what people see when you perform a lecture. And gaining confidence is really important for delivering good performances - instead of thinking about how you're nervous and want to end it as soon as possible, audience will better listen to what you have to say.
Moreover, being stressed harms your health, while doing physical exercises will reduce harm and make use of that adrenaline that your body has generated.
People of mind work often dig themselves too deep in their thoughts, while for interacting with other people you use your physical form - so, when you gain better control of your body, better "know yourself", you will see that it is still frightful to make presentations, but you can handle it. When you know you can handle it, fear will go away eventually.
You might need some backup (cheating notes, e.g.) for in case you get into problems.
Follow some courses, about presentation techniques, public speech,..., and very important, follow courses where you can really test yourself: it makes no sense listening to somebody giving a great presentation if you don't get the opportunity to test yourself. By giving small public explanations about little things, you can get feedback from the rest of the class and you can build further from there.
From the courses, get some small but useful tips and tricks, don't try to implement everything you learn from the course on your first presentation (be aware that it's an evolutionary process, which you'll learn bit by bit).
From my public speech course, I remember that I spoke very fast (I was so anxious that I wanted the presentation to be finished as soon as possible). During some try-outs one time I imagined that all people in the public were foreigners, not native speakers (although this was not the case), and because of that I started talking more slowly and I articulated very good. Since then I'm always using this trick.
Know yourself: every person has his or her own issues while dealing with public speaking: some people don't move and just stand there like a statue, they should move more. Others don't stop moving and they need to be aware not to do that so often. Some people need to hold something into their hands in order to have something to calm their nerves on (I always carry a pen),..., all those things can be detected by yourself and the friendly audience from point 1.
What sets public speaking apart from many other tasks where anxiety could mess up things, is the fact that anxiety is normal here, many people will either feel quite anxious wen giving talks or if they don't feel anything a heart rate monitor would betray that they are under some level of stress. So, the problem isn't so much that you have some level of anxiety, rather that it is too much or your body overreacts causing problems. And then you're bound to get extra anxiety because of the anticipated problems due to anxiety.
The best way to deal with this problem is to do what JamalS says in his answer: you must practice a lot. You then