An Islamic state is a type of government primarily based on the application of sharia (Islamic law).
The early Islamic states were autocracies, caliphates. Today, self-proclaimed Islamic states, such as "Islamic Republics", can incorporate elections and parliaments, however, presumably only as long as these institutions do not contradict sharia laws. One example is the "Islamic Republic of Iran", which has an elected President, but with an Islamic jurist, the "Supreme Leader", having autocratic powers.
See also Islamization and anti-Islamization.