I don't really know how to write emotions and a lot of people seem to have this problem as well, both professional and amateur writers. A fan fic I recently came upon expressed the characters emotions via emoticons, which I found to be very strange. O_O suddenly appearing mid-sentence was jarring.
I've also seen people express the emotions via graphical effects seen in manga and anime. I don't really have anything against this, since they appear to be attempting to maintain the feeling of the source material. The thing is using nosebleeds, sweat drops, cross popping veins, cat smile, chibification, facefaults, waterfalls of tears, etc., are means of showing the characters emotions in a different medium entirely and while it can be done in writing it often does not work all that well. Less so when they attempt to describe the action rather than just saying the effect. Example; Instead of saying something like "Joe facefaulted" they use "Joe fell over onto his face, his limbs in a twisted mass above him".
Now I'll get to the more common means of letting the readers know what emotion the character is feeling. Namely straight out telling us. Susan was afraid. Yoiko was happy. This of course does work as a way of letting the readers know, the thing is it is telling us not showing us what the character is feeling. It also doesn't really draw the reader in and is not really that descriptive. Susan is afraid; okay, how afraid is she - piss her pants scared or just mildly uncomfortable.
The best way, at least I think so, is to describe the effects of the emotion and body language. It is also the most difficult and I'm not good at doing it this way. Example: "Susan's palm's grew sweaty and her heart pounded hard enough that it felt like it would break through her chest." What I was trying to do, I'm unsure how successfully, was to describe some of the physical effects of fear (increased heart rate and force of each beat; increased muscle tension; tremors; sweaty and cold palms; nausea; etc.) instead of saying she was afraid. It is more descriptive than just telling the emotion and hopefully more engaging to the reader.