Shakespearean sonnets were, obviously, named after Shakespeare, who wrote many famous sonnets. The rhyme scheme is as follows: abab cdcd efef gg. There are 3 quatrains and a couplet. A quatrain is a 4-line stanza and a couplet is a 2-line stanza.
A Petrarchan (or Italian sonnet) has a different rhyme scheme that allows for more variation. It is as follows: abba abba for the first 8 lines; the last 6 lines (a sestet) can be any rhyme scheme as long as it does not end in a couplet, but a common rhyme scheme is cdecde.
The rhyme scheme is not as important as the content.
Petrarchan sonnets are modeled off of Petrarch's love poems for a women named Laura. Petrarch never met Laura but he followed her constantly and wrote about her idealized beauty and grace. He also often talked about the pain she made him feel as a result of his love for her. Often he would describe love using elements such as ice or fire as comparisons to love which is generally the opposites, but in this case is a similarity.
Petrarchan Sonnets generally describe women as ideally beautiful (eyes like pearls, lips as red as . . . ) while also using opposing forces such as pain-pleasure/fire-ice to describe love.
Shakespeare wrote some both types of sonnets but the anti-petrarchan sonnets tended to celebrate women for their uniqueness in a realistic way. ie. her skin was not as white as the sand or her eyes were not pearls, but I love her anyway