The festival of Mahashivratri is very important for Shiva devotees. Worshiping Lord Shiva and fasting on this day brings his blessings. The incident of Samudra Manthan is specially mentioned in the story of Mahashivratri.
According to mythology, Samudra Manthan was organized between the gods and demons to obtain Amrit. During the churning, many valuable items came out, but when poison (Halahal) came out of the sea, it was extremely deadly and destructive. The effect of the poison caused a crisis in the whole world.
All the gods and demons were frightened by the wrath of this poison and approached Lord Shiva for a solution. Shiva held the poison in his throat to protect the world. He stopped it in his throat, due to which his throat turned blue and he was called "Neelkanth".
Mahashivratri festival is celebrated for this sacrifice of Lord Shiva and his great work of saving the world. On this day, devotees try to please Lord Shiva by offering water, milk, Belpatra etc. on Shivling.
From this story of Samudra Manthan, we learn that rising above our selfishness and making sacrifices for the welfare of others is a great virtue. This is the true message of Shiva.