A.C.Gimson in his book "An Introduction to the Pronunciation of English" defines a vowel phonetically as, a sound the "articulation of [which] is not accompanied by any closure or narrowing in the speech tract which would prevent the escape of the air stream through the mouth or give rise to audible friction." That is, the air stream from the lungs which articulates the vowel sound flows smoothly and freely out of the mouth without any obstruction or stricture in the mouth parts whatsoever.
He defines the diphthong as "a vocalic element which forms a glide within one syllable." A diphthong is a combination of two vowel sounds. A diphthong has a starting point as one vowel and the tongue glides in the direction of the second vowel. Hence, it's often referred to as a 'vowel glide.'
The answer above has explained the difference between vowel and diphthong very well. Unfortunately, the answer can be misleading about the nature of vowel itself. Answer above has described vowel as a single sound which is true, but it is not any single single sound. It is single sound of a particular type - it is an open sound which is made with free passage of breath. For example the sound of "ah...' a doctor asks you to make when examining your throat. In English language sounds of vowels are represented by the letters a, e, i, o, u, and sometimes w and y.
sound of vowels is distinguished from that of consonants, which are made by organs of speech more or less closed.
Once we understand what is a vowel, a diphthong can simply be defined is sound produced by pronouncing two vowels as a single syllable.
Essentially, a vowel (or more specifically, a monothong) is a single sound (the tongue and lips stay in one place while making the sound) and a dipthong is two sounds blended into one (the mouth has to change shape while making the sound). In fact, the word dipthong means "two tones." Take the words bow and boy for example. Bow has is a monothong sound of a "long o"...if you say it aloud, you'll notice your mouth doesn't change shape while making the vowel sound. Now say boy aloud. Did you notice that your mouth moved? There was a "long o" and a "long e" and they glided together to make one sound...that is a diphthong!