The three main branches of modern Christianity are Catholic, Protestant, and Eastern Orthodox.
The Catholic Church is the oldest; Catholicism dates its founding to the original church founded by Jesus, but reliable historical records only go back as far as the Council of Nicea in 325 AD. Catholicism is at least as old as that, and probably older. Catholicism is the world's largest religious sect, and is particularly prevalent in Latin America. The Catholic Church is headed by a single Pope, who lives in the autonomous region of Vatican City, legally defined as its own country separate from Italy, which completely surrounds it.
The Eastern Orthodox Church emerged in the 4th century and is centered in Eastern Europe, especially Greece and Russia. The Orthodox Church does not have one single leader, but it does have a number of high-level authorities known as Holy Synods. Theologically there are not many differences between the Catholic and Orthodox Church; it's more of a difference in their traditions and rituals.
The Protestant branch is by far the most varied. The first Protestant sect was founded by Martin Luther in 1521 and is still today called Lutheranism. Part of its emergence was due to the rise of the printing press and increased literacy; people could now read the Bible directly and did not have to rely on what priests handed down. Other sects soon emerged in a period known as the Protestant Reformation. Since then, many other Protestant sects have been founded. The Anglican Church is an interesting exception; it was founded before the Reformation, but today is generally considered a Protestant sect all the same. Protestants vary tremendously in their beliefs and practices; their defining attribute is that they assign ultimate authority to no individual person, but rather to the Bible itself.