Personally, I've struggled with performance. I've always thought of music as an intellectual pursuit, and "letting loose" on stage has always been hard for me. I get nervous every time I play for anyone. I'm very critical of myself and assume the audience thinks the worst of me. I slouch, look at my shoes, and frown. I've always admired people who make performing look easy.
These feelings can be even worse when you solo in front of people. Since soloing can be ad-lib, and since you're the center of attention, it can cause even more anxiety.
I've found that to really connect with the audience and put on a good performance, I have to forget myself and really try to have fun. Think happy thoughts like the following:
Sometimes I have to *act* this way with an effort. Internally I'm feeling scared and nervous, but I put on a smile and play TO the audience. It's hard at first, but eventually you realize you are actually having fun!
One famous guitarist, David Leisner, puts it this way:
Enjoy! Don't forget that your performance is the time when you can finally share with your listeners what you have worked so hard in the practice room to achieve. This is a time of joy and not a time for correcting errors or other faults. Players tend to be too self-critical in performance. The practice room is the place for that. The concert hall is the place for celebrating the music. Let your emotions for the music be present. Don't allow minor details to obscure your feelings about the music. Let your excitement for the music be present. Let the adrenaline and your genuine lively passion for the music come through.
My tendency is to beat myself up visibly when I mess up on stage. I'll roll my eyes at myself or shake my head or look at the floor with a red face. This is very bad. When you do this it lets the audience know that you messed up (they probably didn't even notice). Since you're focusing on that, they start to as well, and it takes away from their enjoyment of the music. Messing up also makes me want to play more "right" notes to "make up" for my horrible one wrong note. This leads to a less satisfying solo.
The best way to deal with mistakes, in my opinion, is to laugh at them. Just smile or chuckle to yourself and move on. The few audience members who did notice your mistake will chuckle with you at the inside joke. The people who didn't even hear your mistake will see you smiling at them and smile back.
I put a lot of pressure on myself to solo and play well in front of an audience. Sometimes I'll totally rip it up at home or in band practice and then screw up in a gig because I'm nervous.
Play *simply*. Your audience will be more enthralled with a simple, rhythmic, catchy tune they can whistle than a blazing fast shred-fest with mistakes. Seriously. That doesn't mean you can't get flashy or fast. But play a bit more simply and expressively than you normally do, and you will feel less nervous. Your audience will of course pick up on this and be happy with your laid-back, groovy solo.