What is main reason why Harry and his friends are trying to get the Philosopher’s stone back? Who is this "Philosopher" and who's the stone belong to?

The famed alchemist Nicolas Flamel created the only Philosopher's Stone which (in 1991) was known to exist. Flamel used the Elixir of Life made from the stone to extend his and his wife Perenelle'slifespan for over six centuries.

Harry Potter and his best friends Ron Weasley and Hermione Granger suspected that the stone would be stolen. In overhearing out-of-context conversations, they believed the thief would be Hogwarts Professor Severus Snape.