When speaking of this Book, its correct name is "the First Book of Samuel" (not the book of first samuel).

I Samuel 1:1 - Elkanah (and Samuel) were Levites. For their genealogy, see I Chronicles, chapter 6, esp. v.27-28).
I Samuel 1:3 - Shiloh, half way between Bethel and Shechem on the eastern slopes of the mountain spine of Israel, was one of the religious and administrative centers for early Israel - the Tabernacle and the Ark of the Covenant had been set up there during Joshua's time (Joshua 18:1), and the Israelites met there for major decisions (Joshua, chapters 18, 19, 21, 22:12).
I Samuel 1:4 - Eli was a priest of the Aaronic line, through Aaron's youngest son Ithamar.
His geneaology is not given directly, but can be worked out :
I Samuel 2:27-28 - Eli was descended from Aaron.
I Samuel 4:19-22 - one of Eli's grandsons was Ichabod.
I Samuel 14:3 - the priest at Shiloh during the reign of king Saul was Ahiah, the son of Ahitub who was Ichabod's brother.
1 Samuel 22:9-22 - Ahimelech, son of Ahitub, was priest at Nob until Doeg the Edomite killed him on the orders of king Saul. Ahimelech's son Abiathar escaped and joined David.
I Kings 2:26-27 - king Solomon deposed Abiathar ("of the house of Eli in Shiloh") from the priesthood
I Chronicles 24:1-3 during the reign of king David, one of the priests was Ahimelech the son of Abiathar, one of the "sons of" (descendents of) Ithamar
I Samuel 1:9 - the "temple of the Lord" this was probably a shelter and cult-center for the Ark, built of field-stones rather than a temple in the usual sense. It was not until the reign of king Solomon that the Temple was built in Jerusalem
I Samuel 1:16, 2:12 - "daughter of Belial" and elsewhere in the OT, "sons of Belial" - "belial" meant literally "worthlessness" or "wickedness", and was a word used by the jews to refer to the devil.
I Samuel 2:15-16 - what the priests did was contrary to what was laid down in Leviticus (eg. chapter 3).
I Samuel 2:22 - Eli's sons would seem to have been practising Canaanite fertility rituals and/or wholesale adultery.
I Samuel 3:20 - All Israel "from Dan to Beersheba" - an idiom for the whole of Israel - Dan was the most northerly region, and Beersheba the town in the far south.
I Samuel, chapters 5 & 6 - the Philistines were a maritime people, who probably migrated from the islands of the Aegean (between Greece and what is now Turkey) and settled along the mediterranean coast of the Holy Land. They founded 5 city-states : Ashdod, Ashkelon, and Gaza on the coast, and Ekron and Gath inland. The name "Palestine" come from the Greek word for the Philistines, "palastinoi". Dagon was a sea-god or fish-god ("dag" means "fish" in semitic languages). The plague (hemerrhoids?/tumors?) associated with rodents (6:4) may have been bubonic plague.
I Samuel 6:10-12 - the cows would have been expected to head home for their calves. Bethshemesh was one of the towns granted to the Levites (Joshua 21:16) but at this time it may have had a mixed population of Israelites and Canaanites (Judges 1:33).
I Samuel 7:3-4 - Baalim and Ashteroth - Baalim is the plural of Baal, literal meaning "Lord", used particularly by the Canaanites and Phoenicians as a title for their storm-god. Ashteroth (also plural) was usually a consort of Baal or else a fertility goddess, related to Astarte/Ishtar the goddess of Mesopotamian peoples. It had not taken the Israelites long to become thoroughly syncretistic.
I Samuel 8:1 - Samuel's son Joel - was not the same person as the Prophet Joel.
I Samuel 8:3 - "lucre" means money.
I Samuel 9:9 - at this early date in Israel's history the prophetic ministry was concerned with foretelling future events, "seeing the future", hence the term Seer. Seers apparently worked for money or gifts, as in this case. Samuel was functioning also as a Judge - a religious leader who could guide the people. In later times the Prophets were those who spoke for God to the people.
I Samuel 9:21 - a Benjamite, of the smallest of the tribes of Israel. The Benjamites had been slaughtered by the other Israelites not long before this time (Judges, chapters 20 & 21).
I Samuel 11:1 - the Ammonites lived to the east of the Jordan and at this time were troublesome to the Israelite tribes who had settled there in Gilead and Bashan.
I Samuel 11:5 - in the early days of his reign Saul had not yet set up the attributes of monarchy, but seems to have been behaving more as a Judge of Israel. At this point he takes up the duties of military leader, and in 11:15 Saul is publically proclaimed king and starts his reign.
I Samuel 13:1 - different translations of the Bible have different ways of expressing this verse. The Hebrew literally says "son-of-year Saul at the start of his reign, and two years he reigned over Israel". The way of expressing a person's age in Hebrew is to say "son of so many years", with a number to show the age. This formula for age upon accession, and length of reign is used for the later kings of Judah (eg. II Kings 8:17, 8:26, 14:2, 15:2, 15:33, 18:2, etc.). Saul was obviously not a one-year-old child when he became king, and his reign was much longer than 2 years, so it would appear that the numbers either dropped out of the text or were not recorded. At a rough guess Saul was probably around 40 - his son Jonathon was old enough to be a fighting man and leader of a thousand (I Samuel 13:2), and the age for going to war was reckoned to be 20 (Numbers 1:3) Acts 13:21 indicates that Saul reigned for about 40 years, which is why some versions put his reign at 42 years. The KJV and some other translations give "Saul reigned one year; and when he had reigned two years over Israel . . . . .(he chose 3,000 men)", which does not make particularly good sense either grammatically or logically - he was already leading men in battle at the start of his reign.
I Samuel 13:19-22 - the Philistines had learnt the secrets of iron-working, probably from the Hittites, but made sure that they did not teach the Israelites. 13:21 used to be translated as "they had a file for the mattocks , , " - scholars knew that the sentence dealt with the sharpening of implements, and guessed that the Hebrew word "pim" was some sort of instrument, such as a file, for doing that. However, recent excavations have turned up stone weights (used for weighing silver) - some of which had the name of the weight incised on them - and amongst those was a "pim". So the verse should probably read "They charged a pim for mattocks . . .". A mattock is an agricultural implement with two blades - one at an angle, similar to a hoe, the other like an axehead.
I Samuel 15:21 - much like Aaron with the golden calf, Saul tries to shift the blame : "the people took of the spoil . . .", 15:24 " I feared the people, and obeyed their voice.".
I Samuel 17:4-10 - this reflects an ancient style of fighting, in which armies spend some time shouting insults at each other, untill one side loses self-control and rushes into battle, or else, as here, a duel between champions decides the outcome of the battle (similar confrontations are described in Homer's epic song of the siege of Troy). Goliath was probably about 9 ft. tall, with 120 lbs. of armor.
I Samuel 17:40 - Arab boys (and men) still use pebbles and a sling - though mainly to show their prowess to tourists. They can send a pebble accurately and at very high speed, and "re-load" in a couple of seconds.
I Samuel 17:55-58 - some writers make much of the fact that Saul seems not to recognize David, who had been his musician and armor bearer. However, this enquiry may be more into David's family background, since Saul had promised that whoever killed Goliath should be given Saul's daughter in marriage and have his father's house honored (17:25).
I Samuel 18:10 - Saul "prophesied" in the midst of the house - this may have been an epileptic fit or some instance of ecstatic utterance. Epileptics were often regarded as being possessed by a spirit, either good or bad.
I Samuel 21:1-6 - Ahimelech, priest at Nob, was a great-grandson of Eli. The hallowed bread (shewbread) consisted of 12 loaves which were baked on the Sabbath and set on a table in front of the tabernacle. It was to be eaten only by the priests. (Leviticus 24:5-9)
I Samuel 22:1 - Adullam is south west of Bethlehem. The mountains of the region are of limestone, with many caves where people live even today.
I Samuel 22:3-4 - Jesse, David's father, was the grandson of Boaz and Ruth, who was a Moabite woman.
I Samuel 22:7-8 - by now Saul is a victim of full-blown paranoia, suspicious of his own tribe and his son Jonathon.
I Samuel 24:1-8 - Engedi (Ein Gedi, the "Spring of Gedi") is a beautiful gorge with a stream and waterfalls in a deep cleft between the dry limestone hills on each side. The stream flows from the hills down into the Dead Sea and is one of the very few sources of potable water in the Judaean desert. Today it is kept as a Nature Reserve, and there still are "wild goats" (Oryx) and "rock badgers" (Hyrax) by the sides of the stream, and many limestone caves in the steep sides of the valley. Some of the caves in the area and quite large and run deep underground; archaeological traces show that groups of people lived in some of the caves - for example, at the time of the Roman occupation (ca 63 BC to 150 AD), groups of zealots went underground (literally) and emerged to engage in guerilla warfare against the invaders.
I Samuel 24:3 - Saul went into the cave "to cover his feet" - depending on how earthy your translation is, this may be rendered "to rest" or "to relieve himself"; I might translate it as "to drop his pants".
I Samuel 24:21-22 - David honored this oath when, after the death of Saul and most of his family at the hands of the Philistines (I Samuel 31:1-6), he took care of Jonathon's crippled son Mephibosheth (II Samuel, chapter 9)
I Samuel, chapter 25 - "Nabal" means "stupid", "irreligious", and "ungodly". David and his men had protected Nabal's shepherds from marauders through the year, and now wanted to have a share in the festivites of the shearing. Nabal refused to share food and drink with them, and insulted David to his men
I Samuel 25:34 - literally "any that pisseth against the wall" meant all men and boys old enough to be out of diapers.
I Samuel 25:37-38 - Nabal probably had a stroke, brought on by drunkenness and anger.
I Samuel 26:6-7 - "Ahimelech the Hittite" - the Hittite empire (based in what is now eastern Turkey) had fallen into decay and defeat, and was no longer a great military power. The Hittites no longer went to war with a great army, but the people were still warlike, so individuals or small bands of Hittites would hire themselves out as mercenaries to the surrounding kingdoms. (see also II Samuel 11:6-17, II Kings 7:6)
"Abishai the son of Zeruiah" - Zeruiah was one of David's sisters, so Abishai, Joab (and Asahel - II Samuel 2:18) were David's nephews.(I Chronicles 2:13-17)
"Abner the son of Ner" was the captain of Saul's army, His father, Ner, was Saul's uncle (I Samuel 14:50-51). There was increasing rivalry and animosity between Joab and Abner which culminated in blood-fued and the deaths of Asahel and Abner and (II Samuel 2:18-32, II Samuel 3:20-30).
I Samuel, chapter 27 - David and his men became mercenaries for Achish, the Philistine king of Gath. They are eventually given a town in which to dwell, and turn to raiding some of Israel's enemies while pretending to the king that they are invading Israel
I Samuel 28:4 - "Saul had put away those that had familiar spirits and the wizards, out of the land" - in obedience to Deuteronomy 18:9-14
I Samuel 31:11-13 - Jabesh-Gilead was east of the Jordan, Bethshan west; the towns were about 20 miles apart, so the men of Jabesh had a long trek down into the Jordan valley, across the river, and into the enemy-occupied hills to the West, then all the way back again. One of the first acts of Saul's reign had been to rescue Jabesh from the Ammonites (I Samuel 11:1-11).