This is the approved revision of this page, as well as being the most recent.
Disposing of one's nail clippings
When one cuts (or bites) one's nails, one must make sure that none of the clippings remain on the floor. It’s best to burn one's nail clippings. The next best option is to bury them.  Some poskim allow flushing one's nails down the toilet or washing them down the sink if one leaves the water running for a few minutes. 
When to Cut One's Nails
It is interesting to note that whenever the Arizal would cut his hair he would do so before midday. 
One should arrange for one's hair and nails to be cut on Erev Shabbat. 
Ashkenazim have a practice to cut one's toenails on Thursday and fingernails on Friday or Thursday night. There is also a practice specifically not to cut fingernails on Thursday. Sephardim don't have these practices.
How to Cut One's Nails
Ashkenazim have a practice not to cut one's toenails and fingernails on the same day. Sephardim don't have this practice.
Ashkenazim have a practice not to cut one's nails in order. Rather they should cut them in this order going from left to right (counting from left to right): left hand the 2nd, 4th, 1st, 3rd, 5th, and then the same with the right hand - 2nd, 4th, 1st, 3rd, 5th. See the picture. Sephardim don't have this practice.
↑ Mishna Brurah 260:6 quoting the gemara Moed Katan 18a.
The Gemara Moed Katan 18a writes that if a pregnant woman steps on nail clippings there is a danger that she might lose the child she is carrying. The gemara continues that when one cuts (or bites) one's nails, one must make sure that none of the clippings remain on the floor. It’s best to burn one's nail clippings. The next best option is to bury them. This is brought as halacha by the Mishna Brurah 260:6 and Amot Shel Halacha (by Rabbi Ari Enkin). Rabbi Enkin quoting the Bear Hetiev 260:2 writes that the reason for this is that before the sin of the forbidden fruit, Adam and Eve wore garments made of a fingernail-type substance. After they sinned God took away this Divinely provided clothing. Since it was Eve who was essentially responsible for this loss, our fingernails – which recall these clothes – are said to bring harm to pregnant women. Of course, this is a spiritual and not a scientific matter.
↑Halachically Speaking (vol 4, issue 22) quoting Rav Yisrael Belsky allows flushing one's nails down the sink. Halachically Speaking also extends this to washing the nails down the sink if one leaves the water running for a few minutes. Rabbi Mansour also assumes that this would be acceptable.