The ruler of the Earth at any given time is known as the Manu, the father of mankind (the Manavas). A Kalpa, or day of Brahma, is divided into 14 periods known as Manvantaras (consisting of about 71 Mahayugas each), and during each Manvantara a separate Manu rules. Here is how the Srimad Bhagavatam describes it:
After the end of Brahmā’s night, the creation of the three worlds begins again in the daytime of Brahmā, and they continue to exist through the life durations of fourteen consecutive Manus, or fathers of mankind. Each and every Manu enjoys a life of a little more than seventy-one sets of four millenniums. After the dissolution of each and every Manu, the next Manu comes in order, along with his descendants, who rule over the different [lands]; but the seven famous sages, and demigods like Indra and their followers, such as the Gandharvas, all appear simultaneously with Manu.
In another chapter of the Srimad Bhagavatam, Brahma tells Swayambhuva Manu, the first Manu of the present Kalpa, what his job is:
Since you are my very obedient son, I ask you to beget children qualified like yourself in the womb of your wife. Rule the world in pursuance of the principles of devotional service unto [Vishnu], and thus worship the Lord by performances of yajña. O King, if you can give proper protection to the living beings in the material world, that will be the best service for me. When the Supreme Lord sees you to be a good protector of the conditioned souls, certainly the master of the senses will be very pleased with you.
We are currently living in the seventh Manvantara of the Shwetavaraha Kalpa, and our Manu is Surya's son Vaivasvata Manu, AKA Shraddhadeva, as described in the Srimad Bhagavatam:
The present Manu, who is named Śrāddhadeva, is the son of Vivasvān, the predominating deity on the sun planet. Śrāddhadeva is the seventh Manu. Now please hear from me as I describe his sons.
By the way, Vaivasvata Manu is the son of Surya the son god and his wife Sanjana, whom I discuss in this answer.