Monophone: The pronunciation of a word can be given as a series symbols that correspond to the individual units of sound that make up a word.  These are called 'phonemes' or 'phones'.  A monophone refers to a single phone.

Triphone: A triphone is simply a group of 3 phones in the form "L-X+R" - where the "L" phone  (i.e. the left-hand phone) precedes "X" phone and the "R" phone (i.e. the right-hand phone) follows it.  

Below is an example of the conversion of a monophone declaration of the word "TRANSLATE" to a triphone declaration (the first line shows the "monophone" declaration, and the second line shows the "triphone" declaration):

In the CMU dictionnary, which has close to 130,000 word pronunciations, there are only 43 phones, but there are close to 6000 triphones.