Symbols are objects, characters, figures, and colors used to represent abstract ideas or concepts.

Oathkeeper

The sword Oathkeeper represents honor. It was forged from Eddard Stark’s greatsword, Ice, after it was used to behead Eddard at the order of Joffrey Lannister, and it was given to Jaime by Tywin when Jaime returned to King’s Landing. Ice had been in the Stark family for generations, and so it was an insult to the honor of the Stark family that the Lannisters, after killing Eddard, would melt it down. But Jaime changes the sword’s connotation when he names it Oathkeeper and gives it to Brienne. In essence, Jaime gives Brienne an instrument to carry out her vow to Catelyn Stark that she would find and return Catelyn's daughters. Jaime at the same time upholds his promise to Catelyn that he would send her daughters back if he were returned safely home. In the novel, honor is intrinsically tied to keeping one's oaths and doing what is morally correct, and so the act restores Jaime's honor and allows Brienne to uphold hers. The act also removes the sword's connotation of insult to the Starks as it is now a tool intended to help the Stark girls, so it restores the sword's honor in a sense as well. In these various ways the sword becomes a symbol of honorable behavior.

The Wall

On the one hand, the Wall is a simple defense against the wildlings, but the Wall also acts as a figurative boundary between the natural and supernatural. The Wall exists as a literal defense against the supernatural, specifically the undead creatures called the Others. For the mysterious figure called Cold Hands, who appears to be supernatural, the Wall also presents a literal barrier, as he is unable to pass beneath it after escorting Sam and Gilly south. In fact, the Wall itself even seems to have some magical or supernatural power, evidenced by the fact that Coldhands can't pass it despite there being a gate. Moreover, as Bran's supernatural psychic abilities develop, he feels pulled north toward the Wall, and eventually Coldhands guides him beyond it, suggesting there's some supernatural force on the other side that he must encounter. In the story, it acts as a literal and symbolic boundary between the natural and rational world of the Seven Kingdoms and the magical region north of the Wall full of mysterious and unexplainable forces. (Supernatural events do, of course, take place south of the Wall at times, but they're relatively rare and often disregarded by those who aren't eye witnesses as exaggeration.)

Dragons

Daenerys's three dragons symbolize Daenerys's growing power. The dragons are obviously a powerful weapon in themselves, and it is said in the novel that the Targaryens ruled Westeros for so long specifically because they controlled dragons. But the dragons also lend Daenerys another type of power. Even though they're not yet formidable, being young and relatively small still, because they are the only three dragons in existence people interpret them as a symbol of Daenerys's great destiny. The reasoning among the people that encounter her that's implied in the novel is basically that if she has these rare and terrible creatures, she must have them for a reason. In that way they represent her growing following, and thus her overall power. In much the same way that the dragons are still young and growing, Daenerys's following is also still young and growing, and as the dragons grow and become stronger in the story, so does Daenerys's following. Over the course of the novel, she gains first a small army, then a sizeable force, and finally she rules an entire city. With each step, her power gradually grows, just like her dragons.