The Rema cites on Shulchan Aruch that one should not cut finger nails in the simple order of the fingers, but rather following a particular order, left hand first (Shemiras Shabbos KeHilchoso (42:46). The Arizal disagrees with this practice and others like the Maharam MiRuttenberg were not particular about this. The Aruch HaShulchan (OC260:6) writes that following the Rema does not fall in the category of halachic obligations but a higher level of carefulness. The Mogen Avrohom recommends being stringent. The Chazon Ish cites that this order would not apply to toenails. Overall most Ashkenazim are inclined to this practice so hence this post.

The following handy graphic from kitzur365.org will help to remember this order. Perhaps print this posting out, rip off the Torah portion, and keep this graphic in your bathroom.