Raisins, sultanas and currants are all different types of grape. To be precise, raisins are a general dried grape from the larger grape varieties, and currants are dried smaller grapes of a small, black variety known as the Corinth. Sultanas are a large, seedless variety, and are generally sold dried into raisins.
Blackcurrants are different from currants. A currant (as above) is a type of dried grape, whereas a blackcurrant is a berry that grows on a shrub. Blackcurrants are used to make a certain popular juice drink, and are also eaten on their own. They have a tangy taste.