A hardline Stalinist. A tankie is a member of a communist group or a "fellow traveller" (sympathiser) who believes fully in the political system of the Soviet Union and defends/defended the actions of the Soviet Union and other accredited states (China, Serbia, etc.) to the hilt, even in cases where other communists criticise their policies or actions. For instance, such a person favours overseas interventions by Soviet-style states, defends these regimes when they engage in human rights violations, and wishes to establish a similar system in other countries such as Britain and America.
The term is used to distinguish the rare individuals with these kinds of beliefs from communists more broadly (including Communist Party members), whose adherence to Soviet doctrine and attachment to existing "socialist" states is somewhat weaker.
It is always more-or-less abusive in the sense that those termed tankies do not use the term themselves, but it doesn't have any particular bite (unlike, say, Trot).
The term derives from the fact that the divisions within the communist movement first arose when the Soviet Union sent tanks into communist Hungary in 1956, to crush an attempt to establish an alternative version of communism which was not embraced by the Russians. Most communists outside the eastern bloc opposed this action and criticised the Soviet Union. The "tankies" were those who said "send the tanks in".
The epithet has stuck because tankies also supported "sending the tanks in" in cases such as Czechoslovakia 1968, Afghanistan 1979, Bosnia and Kosovo/a (in the case of the Serbian state), and so on (whereas the rest of the communist movement has gravitated towards anti-militarism).
I wouldn't be surprised if the tankies even defend Saddam Hussein.
Some of the people round George W Bush used to be left-wing, but they haven't really changed their views much; they were mostly tankies.