Those times former US Presidents had to free Americans held by North Korea
The U.S. government warns Americans against traveling to North Korea, specifically stating that travel to the Hermit Kingdom risks "arrest and long-term detention." There have actually been many Americans held by North Korea, most were subsequently sentenced to a fine and years of hard labor. It happened most recently in January 2016, when 21-year-old Otto Warmbier was arrested for stealing a political banner from a hotel. Warmbier was sentenced to fifteen years "for crimes against the state."
Warmbier will likely not be the last American detained by the Democratic People's Republic of Korea (the North's full, ironic moniker). He's actually the thirteenth American to be arrested there since 1996. Most are detained for a number of days but less than a full year. The only exception being Kenneth Bae, who held for nearly two years on charges of attempting to overthrow the government.
Three of those Americans were released only after the interventions of former American presidents acting as diplomats. Since the U.S. does not have official relations with the DPRK (and are still technically at war), the governments cannot speak directly, so when the former presidents flew to Pyongyang, the U.S. government considered their trips "private, humanitarian, and unofficial."
In March 2009, Journalists Laura Ling and Euna Lee were caught and detained for illegally entering North Korea. They were held for 140 days before former President Bill Clinton made an unannounced trip to the North Korean capital to meet with then-President Kim Jong-Il. The two were sentenced to twelve years of hard labor but were released within hours of President Clinton's arrival. The journalists flew back to the U.S. with Clinton.
In 2010, Aijalon Gomes was held for 213 days for illegally entering the country. After Gomes attempted suicide in captivity, an American consular envoy flew to Pyongyang to request the release of Gomes but was unsuccessful. That's when former President Jimmy Carter hopped on a plane and met with Kim Jong-Il in August 2010. Gomes was freed the next day and left the DPRK with Carter.
In addition to Warmbier, North Korea is currently holding Kim Dong-Chul, a naturalized American citizen, for espionage. He has been held for 153 days at this writing.