"Quest" is a possibly somewhat old-fashioned name for an adventure, a journey into the unknown, often involving significant trials and hardships along the way but ending with the acquisition of valuable new knowledge about oneself and/or the world. The quest is often also used as a symbol, representing more than merely a journey for the character(s) involved.
A quest is often a central part of the plot of a literary work. The Knights of the Round Table went on a quest to find the Holy Grail. Symbolically, they were searching for salvation for themselves and a better way of life for their world. In Homer's Odyssey, Odysseus's travels in search of the way to return home is a quest. Again, symbolically his quest addresses his struggles against the Olympic gods.