Grover Cleveland became the 22nd president and served for one term, but was defeated in a reelection bid by Benjamin Harrison, who became our 23rd president. He later defeated the Democratic opponent after Harrison's first term, so therefore becoming the nation's 24th president.
The term for a defeated or term-limited incumbent is a "lame duck" (from its inability to continue on).
lameduck
George W. Bush, Sr. President Bush served his first and only term from 1989 to 1993. He lost his reelection bid in 1992 to Bill Clinton.
He formally announced that he would not seek reelection to the presidency.
President Obama defeated Mitt Romney on November 6, 2012, and was re-elected to a second term as president.
He did not resign but was defeated in his bid for a second term by Ronald Reagan in 1980.
After a closely fought and very expensive race, President Obama defeated his challenger, Mitt Romney, and was elected to a second term as president.
He declared that he would not run for an additional term as president.
Michelle Bachelet cannot be reelected because the Constitution of Chile prohibits the immediate reelection of a president. In other words, Michelle Bachelet could not run for reelection in 2010 because she just finished her term, but she could run for reelection in 2014 after the new president's term is completed.
The designated term of a Confederate president was six years. After that, there was no reelection process. This allowed the Confederate president to not be concerned with the reelection process. It allowed a president to devote a full six years of service without the concern of reelection.
On November 6, 2012, President Obama defeated Mitt Romney and won a second term in office.