But I also don't remember the name of the garment worn by some orthodox jews-- it is a large square cloth, it covers the chest and back, and has a number of fringes (a specific number) on the corners

The one generally worn under your shirt is referred to as a "beged" (beg ed; accent on first syllable) although that's just a generic term for an article of clothing. The fringes are called tsi-tsis. The Torah says they have be attached to the corners of "your four-cornered garment," so they are not actually part of the garment. The word you're probably looking for is "Talit," which a married man wears at morning prayer and on shabbos and which will be his funeral shroud.