I’ve taken ‘unrecognised’ to mean any country that is not recognised by every other country (which expands the list a bit).

• Somaliland, recognised by no other state. Originally British Somaliland, it become independent and merged with Italian Somaliland in 1960. Later it declared independence in 1991 and withdrew from being part of Somalia.

• Republic of Artsakh (Former Nagorno-Karabakh), recognised by three non-UN member states. Declared independence from Azerbaijan in 1992. It is wholly within Azerbaijan, but is not controlled by it.
• Transnistria, recognised by three non-UN member states. Declared independence from Moldova in 1990. It is separated from Moldova by a large river.

• Abkhazia, recognised by 6 UN members. De facto independent of Georgia since 1999, but still claimed as part of Georgia.

• Donetsk People’s Republic and Luhansk People’s Republic(sometimes together called Novorossiya) – two separatist areas in eastern Ukraine, recognised only by South Ossetia.
• Republic of China (Taiwan), recognised by 22 UN members and the Vatican. Claims to be the de jure government of China.

• Kosovo, recognised by 99 UN members. Claimed by Serbia but has been administered by the EU and the UN since the 1990s. Now has a government but it is poorly recognised internationally.

• Northern Cyprus, recognised by Turkey. Illegally declared independence and is claimed by Cyprus.

• Palestine, recognised by 132 UN members and Somaliland. Israel does not recognise Palestine and controls some of Palestine’s claimed territory.
• Sahrawi, recognised by 84 countries and is a member of the African Union. It claims most of Western Sahara, which is controlled by Morocco. Currently governs in exile in Algeria.

• South Ossetia, recognised by 5 UN members. De facto independent of Georgia since 1991, but continues to be claimed as Georgian territory.

• Armenia, not recognised by Pakistan. Pakistan does not recognise Armenia because Armenia supports Nagorno-Karabakh and Pakistan did not because of Pakistan’s support for Azerbaijan.
• Islamic State of Iraq and Syria (ISIS), unrecognised by any country but controls large swathes of territory in northern Iraq and Syria.
• China, not recognised by the 22 countries that recognise Taiwan. The United Nations only accepts the People’s Republic of China as the government of China.
• Cyprus, not recognised by Turkey or Northern Cyprus.
• Israel, 32 countries do not recognise Israel. Israeli territory is also claimed by Palestine, which does recognise Israel.
• South Korea, not recognised by North Korea, claims to be the sole government of Korea.
• North Korea, not recognised by South Korea, claims to be the sole government of Korea.

Bonus:

These are former de jure or de facto countries that still ‘exist’ to an extent. This includes governments in exile and other organisations.

• Sovereign Military Order of the Knights of Malta, does not claim statehood or independence, but has diplomatic relations with 104 other countries.
• Tibet, government is in exile and claims some territory, but has no control over it nor is it recognised by any other nation.
• Islamic Emirate of Afghanistan, Taliban government that claims Afghanistan, currently in exile since 2001.
• Republic of South Maluku, claims part of Indonesia since the 1950s, currently in exile in the Netherlands.
• Biafra, claims to be resurrecting the nation of Biafra, which is currently part of Nigeria. In exile in Washington.
• Republic of Serbian Krajina, currently in exile in Belgrade, claims part of Croatia which it controlled for a short period of time before Croatian independence.
• Federated Shan States, in exile, wants to create an independent state for the Shan ethnic group in Burma.
• Western Kurdistan, wants to create a Kurdish state in Syria, in exile in London.
• East Turkistan, claims independence from Xinjiang in western China, in exile in Washington.
• Chechen Republic of Ichkeria, claimed to be independent by Chechens  government in exile in London.
• Ambazonia, declared independence on 31st December 1999 from Cameroon, currently in exile.
• Koma Civakên Kurdistan, aims to create an independent Kurdish state in Turkey.
• Republic of Cabinda, claims independence from Angola  Currently in exile in the Congo.
• West Papua, claims an independent West Papua, which is currently controlled by Indonesia. In exile in the Netherlands.
• Tamil Eelam, seeks Tamil independence from Sri Lanka, in exile.