Origin
Uzbekistan was a part of the Soviet Union until it became independent in 1991. The communist party changed its name but stayed in power with unchanged leader Islam Karimov. Shortly afterwards Karimov was chosen president of the Republic of Uzbekistan. He remains president up to this day after several elections and referenda extending his term. Severe doubts about the fairness of these elections exist and constitutionally more than two terms for an Uzbeki president are not allowed.  Furthermore human rights organisations are very concerned about the degree of press freedom, freedom of speech and other human rights allowed in this country. 
The repressive character of the political power in Uzbekistan was made most clear in 2005 during what became known as the Andijan massacre. Demonstrations started after the arrest of 23 businessmen in June 2004 charged with extremism, fundamentalism and separatism. During the process more and more people started demonstrating outside the prison walls. In the night from 12 to 13 May 2005 an armed group released the businessmen from prison. Afterwards protests out of general discontent swelled even more. On that same day government troops sealed of the city and started shooting into the crowd. The government claimed a death toll of 173 and blamed the casualties on extremist Islamic groups, with which Uzbekistan indeed has a history. International institutions however established the responsibility of the government for this massacre. Estimates about the number of people killed vary significantly.
Uzbekistan was indeed confronted with Islamic internal conflict in the late nineties and early twenty-first century. Since the early nineties, small extremist Muslim groups were in conflict with the Uzbek state. From the late nineties on the Islamic Movement of Uzbekistan started to threaten the president to step back. Their goal was to turn Uzbekistan into an Islamic state and they carried out several violent attacks to pursue this objective. After attacks on the capital the government started a large counter campaign which meant the end of the movement for now.
Besides internal conflicts Uzbekistan also faces a conflictual relation with its neighbouring countries, which is not violent but continuous up to today. Uzbekistan, as one of the largest cotton producers in the world needs large amounts of water, which it however not unproblematically shares with its neighbours. Many of its neighbours are furthermore dependent of Uzbekistan regarding energy and fuel supplies. It has a large influence on the Uzbek minority in Kyrgyzstan.

Chronology
1991: Independence
1992: Karimov elected president
1999: Islamic Movement of Uzbekistan attempts to assassinate Karimov an threatens to overthrow the government
2000: Violent clashes between muslim groups and the Uzbeki government
March 2004: Attack of islamic groups on the capital Tashkent
June 2004: Arrest of 23 businessmen
May 2005: Protests in Andijan, prison break, Andijan massacre

International community
International human rights organisations and organisations such as the UN and the OSCE have repeatedly uttered concerns regarding the internal (human rights) situation of Uzbekistan.  Also the EU and the US are critical of the Uzbeki human rights record especially after the Andijan massacre. The EU introduced sanctions as from 2005. The US however have a more complicated relationship with Uzbekistan, which is a partner in its war on terror. It shares the Uzbeki governments focus on Islamic extremism in the country.