N.B. I wrote this article in the process of participating in Project Spectra. We originally envisioned including resonance-related exercises at the time, but due to limitations on our access to professional speech therapist input, knowledge, and implementation we only managed to finish pitch-based exercises.
A formant is a concentration of acoustic energy around a particular frequency in the speech wave. There are several formants, each at a different frequency, roughly one in each 1000Hz band for average men. The corresponding range for average women is one formant every 1100Hz. The true range depends on the actual length of the vocal tract. Each formant corresponds to a resonance mode of the vocal tract.
Seen this way, the sound spectra look like mountain landscapes and the formants appear as peaks, a metaphor that is often used for formants. 
The frequency of the first formant is mostly determined by the height of the tongue body:
high F1 = low vowel (i.e., high frequency F1 = low tongue body) low F1 = high vowel (i.e., low frequency F1 = high tongue body)
The frequency of the second formant is mostly determined by the frontness/backness of the tongue body:
high F2 = front vowel low F2 = back vowel
F3: The lower of the formant frequency, the rounder shape of the lip e.g. /U/, /uù/, but F3 is not as frequently used as F1 and F2.
Since the 1950s, and even before, attempts have been made to visualize the vowel space by plotting at least F1 and F2 after collecting large corpuses of data. (Keywords to search for: vowel chart, vowel diagram) One such corpus with 1520 samples of American English was collected by Peterson & Barney in 1952. This particular dataset can be found in various open source libraries such as Praat (http://web.archive.org/web/20190831081419/https://raw.githubusercontent.com/praat/praat/master/dwtools/Table_extensions.cpp) or packages in CRAN.
…, men, women, and children have vocal tracts of markedly different sizes, so that naturally their formants are different. Yet we identify a child’s vowels correctly in spite of this. An /i/ said by a man and an /i/ said by a woman are felt to be “the same sound” and are equated, as far as phonetic quality goes, by the phonetician. On the usual F1/F2 plot they have quite different positions, but on an “articulatory” vowel-diagram they have the same position. Peterson has suggested that a more realistic acoustic diagram is achieved by plotting the ratio of F1 to F3 along the vertical axis and the ratio of F2 to F3 along the horizontal axis, all values being expressed in mels. Then men’s, women’s and children’s “same” vowels are claimed to come out with approximately the same positions.