The United Kingdom
The United Kingdom is one of the biggest and most powerful of the non-Schengen countries in Europe. The United Kingdom has been drifting apart from continental Europe for some time, as evidenced by the United Kingdom's recent exit from the European Union. Some in the United Kingdom want the country to re-establish close ties with continental Europe, including becoming a member of the Schengen agreement. However, with the country still trying to navigate its exit from the European Union, it seems unlikely that it will join the Schengen Area anytime soon.
Cyprus has been a country in turmoil for many decades now. Cyprus is split between those with Greek ties and those with Turkish ties. The northern half of Cyprus is actually under Turkish control, though it claims to be an independent state. The international community (except Turkey) universally recognizes northern Cyprus as illegally occupied Cypriot territory.
Belarus is known for its close ties with Russia, a non-Schengen country, so it is hardly a surprise that Belarus is also a non-member of the Schengen Area. Many former Soviet states, including Belarus, have cold relations with the Western and Central European countries that make up most of the Schengen Area.
Russia is a huge country that is considered to be in both Europe and Asia. Though it is technically in Europe, it has never had close ties with the Central European and Western European countries in the Schengen Area. This only worsened during the Soviet area. In the present day, Russia tends to align itself with former Soviet republics like Belarus. Russia and these former Soviet republics often have agreements to facilitate border crossings that are similar to those found amongst the Schengen countries.
Any country in the Schengen Area must be able to contribute to the Area's border security. Even if Ukraine wanted to join the Schengen member countries, it currently could not because it is unable to control its border with Russia. Also, many Ukrainians feel closer ties to Russia than they do to countries in the Schengen Area.
Albania has serious problems with organized crime. It is common for members of the Albanian mafia to smuggle people, drugs, weapons, and stolen goods across Albanian borders. The government's lack of control over its borders means that Albania is unable to join the Schengen Area.