1. Vowels and Consonants
Phonetically, it is easy to give definitions: a vowel is any sound with no audible noise produced by constriction in the vocal tract, and consonant is a sound with audible noise produced by a constriction.
However, this definition forces us to identify as vowels many sounds which function as consonants in speech. For example, in the English word "yes", the initial [j] is phonetically a vowel according to the definition above. In the phonological system of English, however, the [j] is in a typical consonant position (compare "yes" with "mess", "less", "Tess" etc.). Similarly, there are sounds which are phonetically consonants which under some circumstances do act as syllable nuclei; a typical example would be the use of "syllabic [l]" in English "little" [lItl] (cf. litter).
2. Contoid and Vocoid
A solution to this terminological difficulty, suggested by Pike, is to have two different distinctions, one strictly phonetic and the other based on function, or phonological criteria.
For the phonetic distinction, Pike advocated using the words vocoid and contoid. A vocoid is defined as a "central oral resonant". It's central because not a lateral sound, like [l]; oral because air passes through the oral cavity; and resonant because there is no constriction, so all the sound comes from the resonances in the oral tract resulting from the vibration of the vocal cords. Everything which is not a vocoid is a contoid. Thus, [j] is a vocoid, [i] is a vocoid, [a] is a vocoid, [w] is a vocoid, but [l] is not; it is a contoid, as are [p], [b], etc.
This leaves the terms "vowel" and "consonant" available to be used as phonological terms. Generally, vowels are syllabic vocoids. Thus, of the vocoids above, [i] and [a] could be vowels, but [j] and [w] would not, as they are never syllabic. Consonants are contoids which function as syllable margins, e.g. [p], [b], and sometimes [l] (in words like "lip", "lot", but not the final segment in "little", where the [l] is syllabic.
This definition of vowels and consonants leaves two other possible classifications:
syllabic contoids, such as English syllabic [l] and syllabic [n], or the syllabic fricative [s] in "s'pose", or e.g. syllabic [z] in Chinese [sz] "four".
Consonants and vowels are traditionally classified in two dimensions: place and manner of articulation. Place of articulation refers to the location of the narrowest part of the vocal tract in producing a sound. For example, for the consonant [b] the vocal tract is narrowest at the lips (in fact, it could not possibly any narrower here!). In vowels, the narrowest part of the vocal tract is usually in the middle of the mouth, in the region of the palate. "Manner of articulation" refers to various other things, including whether the airflow is central or lateral, oral or nasal, retroflex or non-retroflex, the phonation type, and the degree of stricture.