A vowel is a sound that is pronounced when the vocal tract is open, as opposed to consonants, where the vocal tract is partially or fully closed. Approximants (glides), such as /w/ and /j/ (English y), blur the boundary between consonants and vowels because they are pronounced with an very small, though open vocal tract as if they were vowels, yet they typically occur at the edges of syllables as if they were consonants, such as in yes and wow.
Vowels in natural human languages can contrast in placement, lip roundness, tenseness/laxness, length, nasalization, tone (and/or pitch accent), and phonation (or voicing). They can also be monophthongs (single vowels) or diphthongs (double vowels - where one vowel or the other generally behaves as a glide). Vowels in nonhuman languages could incorporate any of these features, or yet more, assuming the speakers' anatomy is appropriate.
Placement of vowels is generally discussed in terms of height (or closeness) and backness. For instance, the vowel in the word 'beat,' /i/, can be categorized as a high front vowel; /u/ of 'food' is a high back rounded vowel; and /a/ of 'bark' is a low central vowel.
It is natural in many of the worlds languages to make back vowels round; this creates an acoustic effect that seems to increase the backness, making the two vowels more audibly distinct. In a system like this (as in English), roundness is a feature that is present in the language, but it is not contrastive.
In other languages, like French and German, there are vowels in the same place of articulation whose only difference is the absence or presence of rounding - particularly the front vowels. Compare French dit /di/ (front unrounded), du /dy/ (front rounded), and d'ou /du/ (back rounded). Since rounding increases the 'back sound,' this makes it sound like there are three different places of articulation in terms of back and front (adding a central-sounding vowel), when in reality, there are only two.
In many languages, including English, speakers begin to pronounce nasal sounds (m, n, and ng) before they've finished their vowel; the result is nasalization. However, in some languages like French, Portuguese, and Mohawk, the vowel is nasalized but the nasal sound that originally triggered this pronunciation has been deleted (or was perhaps never there).
In natural languages, vowel systems range from the very simple (i/a/u - e.g. Inuktitut) to the very complex (e.g. English, with ~12 vowels).
What is important to consider in assigning a vowel system to a conlang is the vowel spread. Observe that in most languages with only three vowels, they are spread to the farthest three corners of the chart, forming a triangle. A natural language would never have a sytem of three high front vowels, because speakers want to be able to hear the difference between the vowels as clearly as possible. Even in a language like English, where there are two high front vowels, our chart is still relatively balanced - because we have two of just about every vowel placement, in pairs we call tense/lax (compare /i/ 'beat' (tense) to /ɪ/ 'bit' (lax)).