Lye relaxers contain the active ingredient sodium hydroxide while no lye relaxers contain the active ingredient of a calcium, guanine, or there's one other one hydroxide. Motions relaxer systems are lye, Mizani and Affirm are some examples of some other lye relaxers. Usually the relaxers in a box are no lye ones.
I have used both types in my long history of relaxed hair and I like lye better. Visit Sistaslick's profile. She should have links to her articles that she wrote on the differences between lye and no lye relaxers.