market discount is the amount by which the bond's stated redemption price at maturity exceeds the taxpayer's in the bond immediately after acquisition in the secondary market {IRC 1278(a)(2)}. A taxpayer can choose to accrue the market discount and include it annually in their return as interest income. Otherwise, when the bond is disposed of, gain is ordinary interest income up to the amount of the accrued market discount (a)(1) of the same code section.
Original Issue Discount is when a bond is originally offered for less than the redemption value. for example, if the current rate for a bond was 7% but the issuer only wanted to pay 5% it would issue them for less than par value. a market discount bond also has OID, the market discount is the sum of the bond's issue and the total OID includible in the gross income of all holders before the taxpayer acquired the bond, reduced by the taxpayer's basis in the bond immediately after acquisition The only government OID securities that I'm aware of are the savings bonds and T-bills.
if you want to know more about bond discount see IRS PUBS 550 and 1212
https://www.irs.gov/pub/irs-pdf/p550.pdf
https://www.irs.gov/pub/irs-pdf/p1212.pdf
