Sorry for being arrogant, but I regard myself as a bit of a pro at being calm at comps, having been at the other end of the spectrum a while back.
My first official solve was a 6.38 2x2 LL skip. I was just a little bit nervous

Short version: Rowan's idea that your mindset should be
'lolk solve time now' sums it up perfectly for me! For non-bld events anyway.
I'm mainly bouncing off Chris because he makes lots of interesting points at once - saves me trawling through everybody's!
1) Practice a lot beforehand so that everything about your solving becomes muscle memory. This will help you be able to know what to do even when you're under pressure.
Actually I don't like practising lots before a comp. It makes me care too much. Instead, practise during the months before, and not much in the week leading up to it, just so that you forget how good you are at an event!
3) At the competition on the day of, try not to talk to people between solves. It's nice to talk to people yes, but if you talk to people between solves you lose your focus. Put on your blindfold/earmuffs/headphones etc. to make it obvious that you're concentrating right now and don't want to talk to anyone.
I like having people around who I'm comfortable with, especially the judge. A bit of a light-hearted atmosphere helps me. Before starting a bld attempt I do have a short focus time after the joke or two though.
5) Compete inwardly, NOT outwardly.
Competing inwardly means to compete such as to beat your own personal goals, as they relate to your times only. Try to beat the result you got at the last competition (beat the time you got, NOT your placement in the competition). Try to beat your pb single or average, etc.
Competing outwardly means "I will beat YOU". An example is "I will place in the top 10", or "I will podium", or "I will break the WR". Competing outwardly is VERY stressful, but competing inwardly is not as much.
Yes yes yes! I used to care so much about getting NRs and winning events and stuff. Now I realise that beating people doesn't make me a better person than them, so why am I trying so hard to do so? For me, I prefer to aim to live up to my own expectations. So in 4bld, beating my own WR didn't mean much more to me than getting an OPB, which I knew I could well do. It's just a comp after all, so why not just do what you do the same at home?
One other thought - do lots of events. Keeps your mind off your 'main' events during the rest of the day. Also, if you fail in those, you're bound to have got some randomly nice results in the other events.