When a Jew sees his tzitzis, he is to remember all of Hashem's commandments and should be encouraged to keep them.
How do tzitzis remind us of the mitzvos?
Rashi bases his explanation on a gematria: the word tzitzis (in its Mishnaic spelling, ציצית) has the value 600. Each tassel has eight threads (when doubled over) and five sets of knots, totaling 13. The sum of all numbers is 613, the number of commandments in the Torah. This reflects that donning a garment with tzitzyos reminds its wearer of all Torah commandments, as specified in Parshas Sh'lach 15:39. (the majority of Ashkenazic Eastern European Jews wear Rashi knots.)
Ramban disagrees with Rashi, pointing out that the Biblical spelling of the word tzitzis (ציצת) has a gematria of 590 rather than 600, which upends Rashi's proposed gematria. He posits that in the pasuk, "You shall see it and remember them," the singular form can refer only to the thread of "techeles." The tekheles strand serves this purpose, explains the Talmud, for the blue color of tekheles resembles the ocean, which in turn resembles the sky, which in turn resembles God's holy throne – thus reminding all, of the divine mission to fulfill His commandments. (the majority of Sephardic Jews and Teimani Jews wears Ramban's knots)
Can a Woman Wear Tzitzis?
In Halacha, tzitzis is considered a "time-dependent positive commandment," as the Torah (Shlach 15:39) mentions "seeing" one's tzitzis, and one could not see them in the darkness of night, but rather only in daytime.
In general, women are not required to perform time-dependent positive commandments but may perform them if they choose to.
Therefore, many Rishonim (including Rashi, Rabbeinu Tam, Rambam, and Rashba) permitted women to wear tzitzis. Similarly, the Shulchan Aruch rules that women may wear garments with tzitzis.
Opinions differ on whether women may bless such "optional" commandments; in general, Ashkenazi women do, and Sephardic women do not.
At the same time, other Rishonim, beginning with Maharam M'Rotenburg, hold that women should not wear tzitzis for several reasons.
The Rema states that while women are technically allowed to don a tallis, doing so would appear to be an act of arrogance (yuhara).
The Maharil and the Targum Yonatan Ben Uziel view a garment with tzitzis as a "male garment" and thus forbidden to women as cross-dressing.
Some other sources mention concern for shaatnez or carrying on Shabbas.
The vast majority of contemporary Orthodox authorities forbid women from donning a tallis. However, Rav Moshe Feinstein approved of women wearing tzitzis in private if their motivation is "for God's sake" rather than motivated by external movements such as feminism. (Igros Moshe, Orah Hayyim 4:49)
When the Satmar Rebbe's wife died, she was found to be wearing tzitzis.