Similar to lye relaxers, no-lye relaxers use a product with a higher (more basic) pH to change the shape of the hair strands by breaking those disulfide bonds that hold the protein fibrils. In other words, it's taking out the curl, but also eliminating some of the strength and elasticity of the hair.
Where lye relaxers come in around the 12-14 range, no-lye relaxers have a pH of 9-9.5 and usually use guanidine hydroxide to straighten the hair. Even though the pH is lower with this type of relaxer (making it a "milder" type of chemical to expose a scalp to), they still apply unnaturally high bases to hair. While no-lye is described as gentler, it's not technically true as the hydroxide is what makes the relaxer reactive and both products use this. Furthermore, everyone's hair strands are different and for some, no-lye relaxers actually make hair drier and duller than the lye types. If considering either one, it helps to find a knowledgeable stylist who can examine and help make the decision with you.