The Book of 1 Samuel is about Samuel, Saul, and David. Samuel was the last of the judges over Israel. Saul was the first king of Israel. Much of the book is about how Saul became king, then how he later sinned against God and died. God chose David to be king after Saul. This book shows how God protected David when Saul tried to kill him.
Translators can use the traditional title "1 Samuel" or "First Samuel." Or, translators may consider a clearer title such as "The First Book About Samuel, Saul, and David."
The purpose of 1 Samuel is to give an account of the events leading up to David becoming king of Israel. In 1 Samuel, the twelve tribes of Israel unite under the first king of Israel, Saul. But Saul does not obey Yahweh's commands. So God uses Samuel to choose David as a king who will trust and obey Yahweh.
It is uncertain who wrote 1 Samuel. It may have been someone alive soon after King Solomon died. When Solomon died, the kingdom of Israel split into two kingdoms. The author probably lived in the southern kingdom of Judah. A descendant of David continued to rule over the southern kingdom. The author may have written 1 Samuel to defend David's right to be king. This would prove that David's descendant was the rightful king of God's people.
The writer of 1 Samuel often speaks of "Yahweh's anointed." This refers to the one whom Yahweh chose to be the king of his people. Also, the phrase "Yahweh's anointed" would make later readers think about the future descendant of David who would become king of God's people. This person who Yahweh chose would be known as the Messiah.