Q. What is the reason for the black stripes on a taalis?

A. Ashkenazim maintain the tradition to have black stripes on the taalis as a remembrance to the techeiles that was once a part of the mitzva. Pri Megadim 9: 6, Eishel Avrohom 9-6, See Rambam Hilchos Tzitzis 2-2 that the blood of the chilazon is black, and after it is processed it comes out blue. Sefaradim don’t follow this minhag, and they specifically (for Cabalah reasons) maintain the taalis should be all white.
Sefer Mataamim explains that according to the Zohar (Vaikra p. 227) the black or blue stripes represents the attribute of Hashem’s might (gevurah or din) and the white tallis represents Hashem’s mercy (chesed).
Daas Noite (p. 146) asserts that the stripes are for beautifying purposes.
Others explain that the stripes are used to be able to distinguish if it is already the correct time for reciting a brocho on the taalis, namely, when it is light enough for one to distinguish between white and blue or black techeiles.
Some see in the stripes the reflection of the retzuos of the tefilin. Lastly, a contemporary Magid calls them the bar codes that distinguish Hashem’s nation.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a

Q. Is there a reason for the number and variant width of black stripes on a taalis? Why are they not placed on a cotton taalis koton?

A. See question above where we wrote that some maintain that the black stripes on the taalis represent the retzuos of the tefilin. Therefore some explain that there is seven stripes, since they represent the seven turns of the retzuos there is on the hand, when one wears tefilin. Horav Yaakov Hirshman Shlit’a added that the stripes relate to the seven true ribs of the human body.
Horav Shlomo Miller’s Shlit’a opinion is that the taalis katan since it is usually covered, there may be more leniency in regards to placing the stripes, especially on non-woolen material.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a