Swiss Half Fare Card

Yes.
http://www.sbb.ch/en/leisure-holidays/holidays--short-breaks-in-switzerland/swisstravelsystem/swiss-halffare-card.html

Yes, 1st or 2nd class, you get the same fare as a child, which is exactly 50% of the full fare, except for some local fares under about CHF 5, where the reduced price is about 2/3. On a ticket machine you press the "child / half fare button".
You get the reduced price on trains, buses, trams and boats, anything that moves as shown on the validity map: http://www.sbb.ch/content/dam/sbb/de/pdf/freizeit-ferien/ferien-kurztrips-schweiz/internationale-gaeste/sts-geltungsbereich.pdf
When your ticket is inspected on the train or bus, you must show the Half Fare Card and your ticket. The price is actually CHF 120 for one month, "$126" (you don't say what sort of $) is the approximate conversion.

Thank you, Sam and Chris, for your quick and clear responses.