Lord Hayagreeva is one of the several Avatars of Lord Vishnu. He is worshipped as the God of Knowledge.
Story of Lord Hayagreeva
Narayana took the Avatara of Lord Hayagreeva to return the Vedas to Brahma, and slay the demons Madhu and Kaitabha. He then blesses Saraswathi as the “Goddess of Knowledge”.
Story of the Hayagreeva Idol
After being blessed by Lord Hayagreeva, Saraswathi starts worshipping an idol of Lord Hayagreeva. During Ramanujacharya's visit to Kashmir, Goddess Saraswathi appears in front of Ramanujacharya, gives Him the idol and says that his Bhashyam shall be called as Sri Bhashyam. This idol was passed on from Ramanujacharya to Swami Vedanta Desikar, thorough the Kidambi lineage. Swami Vedanta Desikar was blessed initially with the Hayagreeva Mantra from Garuda, and then received this idol from His cousin. He then gave it to his disciple, Brahmatantra Swatantra Jeeyar, who went on to establish the Brahmatantra Swatantra Parakala Mutt. The idol hence became a part of the Mutt. Even to this day, the idol is worshipped in the Mutt by every pontiff of the Mutt. Special poojas are performed on Sravana star to the idol. Here is a picture of the idol.
Swami Desikan and Lord Hayagreeva
Swami Vedanta Desikar, initially received the Garuda Mantra from his Uncle, and upon reciting the same, received the Hayagreeva Mantra from Garuda. He was then blessed by Lord Hayagreeva himself, and then composed his first work ever, Hayagreeva Stotram. It is a very beautiful piece. You can find the Devanagari script for the same here. Sri Dushyanth Sridhar recently gave the English translation to this work which can be found in this - Dushyanth Sridhar - Hayagriva Stotram. Hayagreeva also helped Swami Desikan in several instances, when Desikan was in need of help.
It is said that worshipping Lord Hayagreeva is very good for students. Reciting the Hayagreeva Stotram, performing Pradakshinam to the Lord etc are said to have helped many.
Information as I have been told from people of Sri Brahmatantra Swatantra Parakala Mutt.