Blue: United States.
Dark Green: Countries currently designated as State Sponsors of Terrorism
Iran: Added in 1984, a few years after the Iranian Revolution and Iranian Hostage Crisis. Reasons include funding terrorist organizations like Hezbollah and Palestinian groups in Gaza like Hamas, Houthi separatists in Yemen, Shia groups in Iraq, providing support to Assad in Syria.
Sudan: Added in 1993, accused of harboring international terrorist individuals and organizations and allowing them to train and operate in the country, most notably Osama bin Laden. In 1996 for example the UN placed sanctions on Sudan for harboring a group that attempted to assassinate Egyptian president Hosni Mubarak, the US accused two Sudanese diplomats of plotting to bomb the UN building in NYC, and of course the infamous 1998 embassy bombings in Kenya and Tanzania which was linked to al-Qaeda elements in Sudan. Also accused of supporting local insurgencies in numerous neighboring countries. The State Department actually notes Sudan tends to be responsive to international pressure and concern about terrorism related to the country, but not enough to be removed from the list.
Syria: Added in 1979 during the designation's inception, Syria's intimate relationship with Hezbollah is a large part of why they are listed as a state sponsor of terrorism. Notably however the State Department says Syria has stopped directly participating in terrorist operations since 1984. They are accused of passive support, such as allowing free flow of insurgents across the Syria/Iraq border in the past, and Syrian officials are suspected of being involved in the assassination of Lebanese Prime Minister Rafik Hariri in 2005.
Light Green: Countries formerly designated as State Sponsors of Terrorism
Cuba: Added in 1982, Cuba was accused of aiding revolutionary movements in a number of Spanish speaking and African countries, the State Department lists support for the ETA in Spain and FARC in Colombia as two reasons for this designation. Cuba was removed from the list in 2015 following the opening up of relations with the United States under the Obama administration, and it has also notably been involved in facilitating peace talks between FARC and the Colombian government which have made a lot of progress recently. This designation has always been controversial as some have claimed it was punitive and politically motivated.
Iraq: Was an OG designated state sponsor of terrorism in 1979 for supporting regional groups to undermine Saddam's foes, but was removed from the list in 1982 during the beginning of the Iran-Iraq war so the US could provide assistance. Was put back on the list in 1990 following Iraq's invasion of Kuwait, and then was removed again in 2004 following the toppling of Saddam's regime.
North Korea: Added in 1988, accused of selling weapons to terrorist groups and harboring members of the Japanese Red Army Faction group, though the primary reason for their addition to the list was the bombing of Korean Air Flight 858 by North Korean operatives. North Korea was removed from the list in 2008 as a result of agreements to reduce NK's nuclear programs, but has largely remained uncooperative in working on terrorism issues, though notably they no longer engage in attacks like the bombing of Korean Air Flight 858.
Libya: Another original member of the list in 1979, sponsored several terrorist organizations that attacked the West, most infamously the 1988 bombing of Pan Am flight 103 over Lockerbie, Sctoland. Gaddafi started temping down terrorism support and dismantled its nuclear program in cooperation with the West so was removed from the list in 2006.
South Yemen: A country that no longer exists but was added on the original list in 1979, not that much information floating around about its inclusion but the gist was it was a Communist state that backed several left wing revolutionary terror groups. It was removed in 1990 when it merged with North Yemen to create the Yemen that exists today.
Sources: State Department site, Council of Foreign Relations, and thismore critical article by Dennis Jett, Professor of International Affairs at Penn State University.