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

II Samuel 1:2-16 - the young Amalekite evidently thought that David would reward him for bringing news of Saul's death. In view of I Samuel 31:4, the Amalekite may well have been lying when he claimed to have killed Saul, probably thinking that that would make him popular with David.
II Samuel 1:18 - "the book of Jasher", (could also be translated as "the straight book" or "the book of the righteous") also referred to in Joshua 10:13, probably a book of records of early Israel. No traces or evidence relating to it have been found yet
II Samuel 5:8 - "Whosoever getteth up to the gutter" (or similar phrases) probably refers to the tunnel and hole through the rock down to the stream in the Kidron valley, which was the water-supply for the inhabitants of Jerusalem. Later on the stream was covered and led into an underground channel built during the reign of king Hezekiah (II Kings 20:20) to enable the city to withstand the siege of Senacherib, king of Assyria (II Chronicles 32:1-22). The water course and the system of tunnels are presently being made into part of an archaeological park in Jerusalem; it is possible to walk along the underground part of the stream (wading along a rocky stream-bed with water several feet deep, while trying to hold a lighted candle in one hand) and see the opening of the hole leading up to what was the Jebusite stronghold. See also I Chronicles 11:6 - Joab was the first man up the water-shaft. The water shaft is now called "Warren's Shaft" after the nineteenth-century archaeologist who rediscovered it and explored the system of underground tunnels.
II Samuel 5:11 - Tyre, on an island just off the mediterranean sea coast north of Israel, was the capital of the Phoenician kingdom. The Phoenicians had lived in the area for hundreds of years; by the time of the early Iron Age (David's time) they had developed into a great sea-faring and trading nation, with colonies all around the Mediterranean. They were Semitic (ie. traditionally descended from Noah's son Shem), and so related to the Israelites, with similar languages and alphabets.
II Samuel 6:3 - according to the instructions given in Numbers, the Levites alone were to carry the Ark (by staves on their shoulders), and only the priests could prepare and cover it before it was carried (Numbers 1:50-53, 3:5-10, 3:30-32 4:5-20).
II Samuel 8:18 - "the Cherethites and the Pelethites" were non-Israelite mercenaries who acted as David's bodyguard. Being non-Israelite, they were less likely to take part in local politics and insurrections. It is thought that the Cherethites came from Crete, as part of the "Sea Peoples' migration" at about or shortly before the time of the Exodus. The Pelethites were probably also part of the Sea Peoples, and are thought to be related to, or a branch of, the Philistines.
II Samuel 12:10-12 - David's bad example was taken up by one of his sons, and resulted in rape and murder within the royal family (chapter 13 and following).
II Samuel 12:31 - what is meant is that David killed the Ammonites in various brutal ways
II Samuel 13:37 - Talmai was Absalom's maternal grandfather. Geshur was a small independent kingdom on the eastern side of Lake Galilee.
II Samuel 15:7 - "after forty years" most probably this is a scribal error, committed at some very early time. The total length of David's reign in Jerusalem was thirty three years (I Kings 2:11), so what is probably meant here is four years.
II Samuel 15:18 - "the Gittites" were from Gath, and were Philistine mercenaries of David's bodyguard.
II Samuel 16:21-22 - a further fulfilment of Nathan's prophecy (II Samuel 12:11). One way in which a king in the Middle East asserted his new reign was to take over the harem of the previous king. Absalom's dalliance gave David time to escape and regroup his forces.
II Samuel 17:1-14 - Hushai counteracts Ahithophel's good advice, and gains further time for David.
II Samuel 20:13 - Amasa had been the general of Absalom's army. He was also another of David's nephews. His appointment as general in Joab's place led to his murder by Joab (II Samuel 20:8-10).
II Samuel 21:1-9 - see Joshua 9:3-27 & I Samuel 20:14-17.
II Samuel 21:16, 18, 20, 22 - "the sons of the giant" meant sons of Goliath.
II Samuel, chapter 22 - compare Psalm 18.
II Samuel 24:1-4 - the Bible does not give us the reason why David wanted a census of the people, but it could have been for tax purposes or for gauging the potential size of his army, either of which might indicate reliance on one's own strength rather than on God's.
II Samuel 24:16-25 - the threshing floor of Araunah would be a bare rocky patch where the grain could be swept up from the floor. Araunah's threshing floor is believed to be where the Temple was eventually built, which was indeed on a rocky hilltop overlooking the "City of David". This rock is now the central part of "The Dome of the Rock", one of Islam's most holy places, on the Temple Mount in Jerusalem.