How the yield is found
Yield to maturity is the one annual rate that makes the present value of every remaining coupon, plus the face value at maturity, equal to the price paid. There is no direct formula for it, so the calculator brackets the answer and closes in with Newton steps until the price matches.
Dirty price = Σ C ÷ (1 + y/f)^t + 100 ÷ (1 + y/f)^TC = coupon per period per 100 of face, f = coupons a year, t = periods until each coupon, T = periods until maturity.
Clean price, dirty price and accrued interest
Bonds are quoted at a clean price. Between coupon dates the seller has already earned part of the next coupon, the accrued interest, so the buyer pays the dirty price: clean price plus accrued interest. Choose "By dates" to count the days: actual/actual, as U.S. Treasuries do, or 30/360, which MSRB Rule G-33 sets for municipal securities.
Yield to call
For a callable bond the same solve runs to the call date, with the call price in place of face value; the lower of the two yields is the yield to worst.
Example: 10 years left, a 4.5% semiannual coupon, price 96.25. Twenty coupons of 2.25 and the final 100, discounted at 4.981% a year (2.490% per half-year), come back to 96.25.