A monophthong is a single vowel sound - the monophthongs used in English, with examples of the words containing them, are :
/æ/ hat
/i:/ read
/ɪ/ ship
/ʊ/ book
/uː/ too
/e/ bed
/ə/ reader
/ɜ/ː bird
/ɔː/ bore
/ʌ/ luck
/ɑː/ card
/ɒ/ cod
A diphthong is a vowel sound which, within a specific language, is usually considered to be a "single" sound, but actually consists of two vowel sounds occurring consecutively in the same syllable, which "glide" into each other. In English, the diphthongs used are :
/ɪə/ near
/eɪ/ play
/ʊə/ pure
/ɔɪ/ boy
/əʊ/ go
/eə/ bear
/aɪ/ high
/aʊ/ how
A triphthong is a consecutive sequence of three vowel sounds which glide into each other. In British English, the following triphthongs are found :
/eɪə/ player
/aɪə/ higher
/ɔɪə/ oil
/əʊə/ lower
/aʊə/ our
In many varieties of English triphthongs tend to be avoided by the use of an "intrusive" consonant - eg our will be pronounced /aʊwə/ rather than /aʊə/ and player as /pleɪjə/ rather than /pleɪə/. However, in other varieties of English, eg Australian English, they are common.
See also : Vowel sounds
Further Reading
Roach, P. English Phonetics and Phonology CUP