A bond is an investment that represents a loan. They're typically issued by governments and corporations who want to borrow money. A borrower who issues the bond promises to pay its lender, the bondholder.
The price of a bond and its interest rate are inversely correlated. That's because a higher interest rate makes bonds more attractive to lenders and less attractive to borrowers. Higher demand and lower supply means higher prices. Lower yields make bonds less attractive to lenders and more attractive to borrowers. Lower demand and higher supply means lower prices.
How inflation expectations affect demand for bonds
Generally speaking, bond investors are promised a fixed amount of money in non-inflation-adjusted currency. The more inflation, the less valuable their future payments become. The less inflation, the more valuable (relatively speaking) their payments.
When inflation expectations rise, therefore, investors demand a higher interest rate for their investment as compensation for that lost value, other things being equal. Demand for bonds falls, bond prices fall, and interest rates rise. When inflation expectations decline, investors will be more willing to lend money. Demand rises, bond prices rise, and interest rates fall.
How inflation expectations affect the supply of bonds
Of course, borrowers would prefer to repay their debt with future money that's less valuable than the money they borrowed in the past.
Higher inflation expectations will therefore make them more willing to borrow money. Supply should increase, bond prices fall, and interest rates increase. Lower inflation expectations make borrowers less interested in issuing bonds. Supply decreases, bond prices rise, and interest rates decrease.
Putting it all together...
Higher inflation expectations decrease demand for bonds and increase their supply. Both factors result in lower bond prices and higher interest rates.
Lower inflation expectations increase demand for bonds and decrease their supply. Both factors result in higher bond prices and lower interest rates.
Of course, inflation expectations can have other effects on the economy -- over Federal Reserve interest rate policy, economic growth, and employment, and so forth. These factors can have their own effects on interest rates.