The first three systems (of five) help us out by giving away their identities in their names. You will see what I mean:
Pure accentual verse counts only the stresses, or accents, in a line. (This system also goes by the "strong-stress system.") The oldest poetry in English is generally pure accentual verse, and we see such verse now in nursery rhymes: in the lines, "Hickory dickory dock, / The mouse ran up the clock," the lines need to have three stresses to sound right, but the number of unstressed syllables can shift. For instance, the third line of that rhyme, "The clock struck one, and down he run," has one fewer syllable than the sixth, "The clock struck two, and away she flew," but the stress pattern remains the same, so both lines work in this system.
Accentual-Syllabic verse, on the other hand, counts the number of syllabes in every line as well as the number of stresses. The vast majority of poetry in English until recent times falls into this category. The basic form of the famous iambic pentameter for instance, has ten syllables per line, five of which are stressed. (For more details, see the Feet and Lines sections.) If a poem follows rules that govern the number of stresses and syllables in each line, it is accentual-syllabic verse.
Pure Syllabic verse, as you might suspect by now, counts only the number of syllables in a line, regardless of the number of stresses. French and Japanese poets have written pure Syllabic verse for centuries, but poets writing in English have seldom done so until recently, and even then rarely. Most people know of pure syllabic verse through English-language imitations or translations of haiku, with five, seven, and five syllables in the three lines of a poem. (Strictly speaking, a haiku is more than a poem with a certain number of syllables per line, but that is a chapter in another story.)