According to the Scientific American, during a nuclear meltdown, core material overheats, either due to a runaway reaction or loss of coolant. The extreme temperatures can cause damage to the reactor, partially melt the nuclear fuel and contaminate any remaining coolant with radioactive material. In addition, some reactor designs use pressure vessels, and a meltdown can increase pressure enough to cause an explosion or release radioactivity from the containment vessel.
The Scientific American explains that one of the major dangers of a nuclear meltdown is the possibility of a runaway reaction. Under normal circumstances, the core material is kept in contact with neutron-absorbing control rods in order to slow down or stop the reaction, as necessary. A meltdown can damage the control mechanism, preventing these rods from engaging, or it can melt enough nuclear fuel together to create a self-sustaining reaction. In these cases, the fuel can continue to produce heat and dangerous radioactivity long after the initial accident ends.
Wikipedia notes that the most infamous nuclear meltdown occurred on April 26, 1986, when the Chernobyl Nuclear Power Plant suffered a catastrophic failure. A coolant failure led the core to build up heat to dangerous levels, creating a steam explosion that blew off the roof of the reactor. The explosion spread radioactive material over a large area, contaminating the surrounding city and countryside for decades.