Please see our site's article on the Fertility Awareness Method.
The halachic factors that go into assessing whether contraception is permissible apply equally to natural family planning. The fact that it is "natural" does not change the fact that it is a form of contraception.
The halachic advantage of natural family planning is that it does not entail questions about its mechanism – selective abstinence – which is natural and indirect. However, since this technique is less efficacious than others, there is a halachic tendency to prefer other methods when there is a strong reason to avoid pregnancy.
We refer you also to the discussion of contraception in general, and the rhythm method in particular, in Dr Avraham Steinberg's Encyclopedia of Jewish Medical Ethics. (This link leads to an abridged version of the article, in Hebrew).
The Lactational Amenorrhea Method raises no halachic concerns at all, since it relies on breastfeeding rather than on abstinence. However, it is effective for no more than six months postpartum.
Please get back to us with any further questions.