The first water, Matsya, was taken by Lord Vishnu at the end of the Satyuga (first age), when a flood destroyed the world. Through this avatar, he saved humanity and the sacred vedic texts from the flood.
The second avatar, Kurma, was taken in the Satyuga (first age) to help the Devas and to obtain the amrit (nectar) of immortality which was also sought after by the Asuras (demons). He helped in creating the world by giving support on his back through this avatar.
The third avatar, Varaha, was taken at the end of the last flood in Satyuga (first age), when Bhoomi Devi (Earth Mother) sank to the bottom of the ocean. Lord Vishnu, in the form of Varaha, dived into the ocean and raised the Goddess out of the ocean, supported by his two tusks.
The fourth avatar, Narsimha, was taken in Satyuga (first age) to kill a tyrant demon king. Narsimha is the only avatar which was hybrid in form being half human and half animal.
The fifth avatar, Vamana, appeared in the Tretayuga (second age) in order to destroy Bali, the king of demons. He came during a huge ceremony conducted by the king and cleverly asked for just three feet of land, measured by his own small feet. Lord Vamana covered whole of the earth and the heaven, subduing Bali into his feet.
The Sixth avatar, Parshurama, appeared in the Tretayuga (second age) to destroy the warrior caste. When the king of the earth became despotic and started to harm people and saints in the forest, Lord Vishnu incarnated as Parshurama and destroyed all the kings who were harassing the people.
The seventh avatar, Lord Rama, the prince and king of Ayodhaya, appeared in the Tretayuga (second age), to kill the demon Ravana. Lord Rama is one of the most popular avatar of Lord Vishnu . He is known as "Maryada Purshotam", an "Ideal Man". His life story on earth was written by sage Valmiki in Sanskrit and later in Hindi by sage Tulsi Das.
The eighth avatar, Krishna, along with his brother Balarama, appeared in the Dwaparyuga (third age) to kill the demon king, Kansa. Lord Krishna conveyed the message of love and humanity to the world.Lord Krishna told the epic poem Bhagavad Gita to the warrior Arjuna ,in which he acclaimed : ''Whenever Dharma, or the situation of law and order, is endangered on this world, I incarnate onto this world to re-establish Dharma, law and order, and to protect the Sadhus or saints and to destroy the evil elements of the society.''
The ninth avatar, Mahavira Buddha, appeared in the Kalyuga (fourth age) , to teach the lesson of following a middle path in life. ''Buddha'' means 'the enlightened one'. He taught that all sorrow comes from attachments and desires, so it's better to curb all attachments in order to remain happy.
The tenth Avatar, Kalki, ("Eternity", or "time", or "The Destroyer of foulness"), is yet to appear on the earth. And it is expected to appear at the end of Kalyuga (fourth age), the time period in which we currently exist, which will end in the year 428,899 CE.