Time for a quick biology lesson: IUDs (intrauterine devices) primarily work as both ovicides and spermicides though there are some who believe that there *may* be some secondary effect on already implanted embryos and this is where the objection lies. Similar with the combined oral contraceptive pill, the primary function is the prevention of ovulation though it *could* have an effect on implanted embryos. Hobby Lobby regards these types of contraceptives to be abortifacients and therefore contrary to their 'sincerely-held religious beliefs'.
Where their argument falls apart is here: The American College of Obstetricians and Gynecologists in 1972 adopted the definition of conception as: 'Conception is the implantation of the blastocyst'. Or put in layman's terms: pregnancy doesn't begin until implantation occurs. This was backed up in 2004 by the American Medical Association that hormonal contraception can "affect implantation but not terminate an established pregnancy". So the main ways that contraceptives work (preventing ovulation, preventing fertilization, and preventing implantation) are all prior to the start of a pregnancy and therefore, non-abortifacient.
But there are lots of contraceptives that their health plan does include, why can't those women just pick one of those other types? Because not all contraceptives work for all women and contraceptives are used for reasons other than pregnancy prevention: treatment of severe menstrual cramps, polycystic ovary syndrome, and endometriosis, just to name a few. Most importantly, those are decisions to be made by a woman and her doctor as to what is the best health care for her. No one else should be involved in that conversation. No one. Ever. It's telling that the very folks who want government in as little as possible are first in line to want to peer into the bedrooms and uteri of Americans. Corporations and government have no business being part of that conversation; particularly corporations that purchase products from China (Hobby Lobby) where abortion is not only legal, but also enforced and that invest in companies as part of their 401 (k) offerings that manufacture the very products they refuse to cover for their employees (again Hobby Lobby).
Beyond that, this decision opens a huge can of worms: if 'closely held' corporations whose owners have sincere religious beliefs, even though the majority opinion didn't exempt companies from having to offer vaccines (Mennonites), blood transfusions (Jehovah's Witnesses) and certain products derived from pigs (Muslims), who's to say that those challenges couldn't be mounted in the future? The precedent is now there and could be applied to virtually anything. And as it has been estimated that between 52 and 90% of companies in the US are 'closely held'; that's a huge number of people who could be affected by the religious beliefs of a few.
Now THAT is a true violation of the First Amendment.