The repetition of consonant sounds at the beginning of closely connected words, as in 'Peter Piper picked a peck of pickled peppers,' is known as

Alliteration is the repetition of the same initial consonant sound in a series of words in close proximity. The repeated 'p' sound in the example is a textbook illustration of alliteration. Assonance involves repeated vowel sounds; consonance involves repeated consonant sounds anywhere in words (not just at the start); onomatopoeia refers to words that imitate sounds.