Two ways are common. The basic way (since at least the 9th century) has been to scoop a small amount of the Body and Blood on a golden spoon and place it in the communicant's mouth (in some Churches, you don't close your mouth until the spoon is removed, in others you do).
In the 19th century, the Melkite Greek Catholics, inspired by French dread of les microbes, began cutting the prosphora up into julienne strips, dipping a particle into the Chalice before placing it in the mouth of the communicant. However, this is very much an innovation, and not particularly better from an hygienic perspective than using the spoon. Nobody has ever gotten ill from receiving communion with the spoon, to the best of my knowledge. Such concerns are overblown, if not entirely misplaced.
As to why different modes of communion emerged, the oldest stratum of the Tradition, both East and West, has the faithful receiving each element separately, the Body in the hands, the Blood from the Chalice. Gradually, this method became reserved for the clergy out of a fear of profanation, when the massive influx of converts of dubious catechesis greatly increased the number of people receiving.
In the West, the Body was placed on the tongue of the communicant, but until the 13th century, he still received directly from the Chalice, a practice not restored until Vatican II.
In the East, by the 9th century the Body and Blood were mixed together in the Chalice, though precisely why it was decided to do so is not clear.
