Lots of info here tonight...we'll brief each chapter.
Chapter 21: David begins hiding from Saul at Nob. He lied and told the people that Saul sent him there, so that he could safely hide. He took the sword of Goliath while he was there. Then he went to the king of Gath who said, "Aren't you the mighty David?" and then David began to act crazy in order to keep his cover so that they wouldn't kill him.
Chapter 22: David acquires his own followers/army. He took his parents to be safe in Moab at this point, but the prophet Gad told him he wasn't safe and should return to Judah. David then goes back to Judah. Saul discovers David's whereabouts and he and his army go to Nob and slay everyone and the one man left, Abiathar, ran to David--who then takes the blame for the death of this man's family--who promises to keep him safe.
Chapter 23: David takes his men and goes to Keilah, who were fighting the Philistines, and helps them fight the Philistines. Saul again discovers David's whereabouts, but David speaks with God who tells him that the Keilahites will turn him over to Saul, so David runs off into the wilderness and meets up with Jonathon once again. Jonathon tells David not to worry and that he will someday be the rightful king of Israel. The Ziphites (where David was hiding) told Saul all of the possible hiding places in the wilderness, but Saul then got called back to Israel as the Philistines were attacking once again.
Chapter 24: After fighting the Philistines, Saul hunts David down to a cave in Engedi. However, David is not found and therefore David is given the chance to take Saul's life, but instead just cuts his skirt. After that skirmish, David says, "Hey, Saul! I could have killed you, but I chose not to kill the Lord's anointed because what kind of king would that make me?" Saul then says, "You are such a good person--you are the rightful successor to the throne. When you become king, please spare my family." (Side note: at this time, it was customary for the current king to kill the former king's family.)
Chapter 25: Samuel dies. David then goes to Carmel and asks Nabal, a wealthy man, for some help and protection from Saul, to which Nabal says, "Get bent! Why should I help every servant who leaves his master!" So David and his men get ready to smite Nabal and his servants and family, but Nabal's wife Abigail gathers a peace offering and says, "Please, spare us. My husband is an idiot. Consider me your handmaid...but I won't tell Nabal." So Nabal gets plastered and in the morning after he sobers up, Abigail tells him everything that happened and Nabal gets so upset that "his heart died within him and he became as a stone." Nabal dies 10 days later and David rejoices about this and then takes Abigail as his wife...and picks up another wife along the way.
Chapter 26: Another game of hide-and-seek with Saul. Saul finds David again and sets up camp. Saul and his men sleep (probably in preparation for the big kill) and David and his men find Saul and take his sword and his canteen. When Saul wakes up, David calls to him again and says, "I had yet another chance to kill you and yet I spared your life." So Saul says, "You really are good. You will be king one day. Come back with me." But instead they part ways.
Chapter 27: David wants to find a new hiding place from Saul, because, lets be honest, Saul hasn't exactly been a man of his word to this point. So, David takes his army of 600 and his 2 wives and goes to Gath to hang out with the Philistine king Achish. David goes into battle and expands the lands of Achish. Achish is happy with what David is doing for him and is convinced that the people of Israel will now hate David.