How does the treasury determine the rates of the bonds they sell? My understanding is that price and yields are determined via supply and demand. How are the interest rates determined when the bonds are auctioned off?
The question appears to refer to what is normally called the coupon rate (but the US Treasury does call the interest rate in its documentation). This is different than the yield, which is determined by a 1:1 function of price.
Issuance conventions vary across markets. I will just use the U.S. Treasury as an example. Firstly, this only applies to new bonds/bills, since reopenings already have a coupon, and bills are discount instruments (no coupon). I will ignore inflation-indexed bonds.
For the auctions inquestion, competitive bidders specify yields to 3 decimal places. Then:


Single-price auctions. The interest rate we establish produces the price closest to, but not above, par when evaluated at the yield of awards to suc- cessful competitive bidders.


That is, the coupon rate is set to get a price closest to par from below, using 1/8% increments.
Quote from: section 356.17 oflink to regulations
Treasury auction information page.