The four ages
Hindu cosmology describes four ages in a great cycle: the Satya or Krita Yuga, the age of truth, when dharma stands on four legs; the Treta, when it stands on three; the Dvapara, on two; and the Kali Yuga, the present age, on one. Their lengths are in the ratio 4:3:2:1, and together they make a Mahayuga of 4,320,000 years.
The days
Two of the Yugadi days are well known as festivals of undecaying merit: Akshaya Tritiya in Vaishakha, the start of the Treta Yuga, and Akshaya or Amla Navami in Kartika, the start of the Satya Yuga. The start of the Dvapara Yuga is placed on the Trayodashi of the dark half of Bhadrapada (in the Purnimanta reckoning), and the start of the Kali Yuga on the new Moon of Magha, Mauni Amavasya, a great day for bathing at Prayagraj.
Some texts give slightly different tithis, and some reckon the yugas from other days; the list above is the one most often followed.
How they are kept
Bathing in a holy river, charity, fasting, the worship of Vishnu and offerings to the ancestors are praised on all four days. The merit of charity on them is described as akshaya, inexhaustible.
Yugadi and Ugadi
The name is shared with Ugadi, the Telugu and Kannada new year on Chaitra Shukla Pratipada, which is also called Yugadi, the beginning of an age, since Brahma is said to have begun creation on it. That day is a new year festival, not one of the four tithis above.