The Gemorah says that if a pregnant lady steps on nails after being cut, she may loose her child that she is carrying. One of the reasons given is, before Adom sinned his body was covered with a layer of nails. After he sinned the nails only remained on top of the fingers. Since a lady caused the sin of the eitz hadas that in turned caused the removal of nails fromthe body, women are punished because of it. Based on the aforementioned if one burns the nails he is considered a chassid. If the nails are buried he is considered a tzaddik, and if the nails are thrown in a place where they will be stepped on he is a wicked person. Men should also refrain from stepping on nails. Toenails should also not be thrown and they should be dealt with like fingernails. Steeping on nails of a non-Jew should no be done.
The Pri Megadim (OC 260) says: Adam HaRishon was created covered by nail like material over his skin. After the sin it was removed, remaining only on his fingers and toes. Since Woman is generally blamed for the original sin, there is therefore a danger for pregnant women to step on finger and toe nails.
Some other rules of (thumb)nails:
Not to cut on Thursday because they will sprout (presumably this means noticeably) on Shabbos.
Not to cut finger and toe nails on the same day (Mogen Avrohom 260). (Shulchon Hatahor recommends averting this problem by leaving one toenail uncut).
Only cut fingernails Erev Shabbos and Yom Tov (Ibid).
Based on the above the Mogen Avrohom recommends cutting toenails on Thursday and fingernails on Friday.
The Rema says not to cut fingernails in order. The order he gives is 42531 for the left hand and 24135 for the right. The Arizal laughed at this practice and the Maharam Miruttenberg was not particular about this. The Mogen Avrohom recommends being stringent. There are various opinions as to which hand to cut first, the Rema seems to favor the left. (See Pri Megadim and Ashel Avrohom Mibutshetsh). All this does not apply to toenails. (Chazon Ish).