Two live disagreements attach to the tallit katan, and both are about how literally to take the verse behind it.
The first is visibility. Numbers 15 says of the fringes that you shall see it and remember all the commandments, and one reading takes the seeing seriously: the fringes should hang outside the clothing where they can actually be seen. Many Hasidic and Yeshivish communities wear them out for that reason, sometimes with the whole garment visible over the shirt. Another reading holds that the verse describes the purpose rather than imposing a manner, that the wearer knows the fringes are there, and that in a society where displaying them invites hostility or ridicule discretion is preferable. Modern Orthodox practice frequently tucks them in, and Sephardi custom in many communities does the same. Neither side treats the other as failing the commandment.
The second is material. The obligation in the Torah is stated of a garment, and the surrounding passage in Deuteronomy speaks of wool and linen. A body of halakhic opinion, including that associated with the Vilna Gaon, holds that the biblical obligation properly attaches only to a woollen garment, and that wearing a cotton tallit katan fulfils at best a rabbinic obligation, so that a person who wears cotton and recites the blessing may be blessing over nothing. Those who follow this wear wool year round, including in hot climates, and it is one of the more visible instances of a stringency adopted at real cost.
The garment itself is a modern solution to an old problem. The commandment attaches to four cornered garments, ordinary clothing stopped having four corners, and rather than let the commandment lapse a garment was devised whose only purpose is to supply them. It is also called arba kanfot, four corners.