What is Shoivavim?What is Shovavim Tat? what is its significance and how is it for lack of a better term celebrated?
Sources Shaar HaKavanot Drushei HaLaila
Sha'ar Ruah Hakodesh Tikun 27 page 17c
From Devarim Shebikedusha Chapter 4 in Hok L'Yisrael- in fact the musar section of Hok L'Yisrael for the entire six weeks typically deal with the Shovvavim):
There is also the Sefer Beniyahu BenYehoida which deals with the primary tikkunim of this period of time. There are varying minhagim of the Mekubalim during this time. Many Mekubalim fast day and night Shavua L'Shavua, while others will fast 2-3 days(day and night) each week, based on the Arizal's teaching that 2 days(day and night) are equal to 27days of fasting, and 3 days(day and night) are equal to forty days(see Sha'ar Ruah HaKodesh Tikkun 1 page 6b, introduction to Sefer Beniyahu BenYehoida and Evn HaShoham hilchot teshuva siman 1).
Also many of the Kabbalistic Yeshivot in Jerusalem host tikkunim on Mondays and Thursdays in which, following the teachings of the RaShaSh and the Ben Ish Hai(see Lashon Hakhamim) the various fasts can be redeemed through tzedaka for those who are unable to fast.
is the initials of the parshas:
Shmos Vaeira Bo Beshalach Yisro Mishpatim (then optionally, Terumah Tetzaveh). So that makes it a period of 6 (or 8) weeks in the wintertime. It generally fills the a gap in the Jewish calendar post-Chanukah, pre-Purim. Of course the word "shovevim" means "wayward", as in "come back, wayward children!"
It's traditionally considered a time (especially in Hassidic communities) to focus on the observance of commandments related to marital intimacy.
In many communities across the gamut of Orthodoxy, refresher/review courses in taharat hamishpacha (family purity) are offered during this time.
I don't know what the oldest source for the backgrounds of "shovevim" are, but that's how it's done today.
In the Hassidic community of Satmar, it is a time for direct confrontation with one's baser desires, and sermons during this period will bring this up explicitly. There's the story of some fellows from the Skver Hassidic community (whose attitude is to focus on other things instead) going to Satmar during this period and hearing what was to them quite a shocking earful!