MAO metabolizes serotonin, norepinephrine and dopamine. By inhibiting this, MAOIs increase levels of those neurotransmitters. Tyramine is an amino acid which is normally metabolized by MAO in the gut. This is bad due to tyramine's effect on norepinephrine and leads to a hypertensive crisis. Sometimes a Ca-blocker like nifedipine is prescribed for the user of the MAOI to take to combat this effect should it occur.
Tyramine is in cheese, wines, and anything aged or fermented.
People have a tendency to confuse these two categories. Just a note to add to the very clear explanation from Lamont Granquist. Some drugs such as the South American tribespeople's Ayahuasca or Yage contain a mixture of two plants, one containing DMT and the other the inhibitor which prevents the DMT from being metabolised. And I have heard that seesions involving just that mixture have been on offer in California. Updated 1994, please send me any new information.