I am only going to include characters that we have seen in the books, so no Arthur Dayne, Aemon the Dragonknight, etc., and I won't take into account people's prior conditions; again, only what we have seen in the books, so no fit Robert or young Barristan.
Top Tier: Tie
Sandor Clegane: The Hound is certainly one of the best warriors seen in the books. He is a huge man, yet one of the quickest swordsman in the realm. He has demonstrated his ability to defeat larger opponents (Ned had the impression that the Hound was intentionally not killing the Mountain during their fight at the Hand's Tourney) and to defeat groups (both at the tavern and outside the Twins).
Jaime Lannister: The Kingslayer is, without a doubt, the most technically skilled swordsman in the books other than Barristan, who is old, and Syrio, whose style is impractical when facing an armoured knight. He may have been no more than an average commander, but Jaime still nearly ended Robb's Rebellion singlehandedly at the Whispering Woods, only failing because his sword became stuck in the body of a Karstark he killed. The only reason I do not place him in the top tier alone is due to the Hound's significant physical advantage and greater ability to fight through wounds.
Second Tier
1. Garlan Tyrell: Although Loras is the more acclaimed Tyrell fighter, he himself admits that Garlan is better with a sword. Garlan Tyrell routinely trains against three or four men at once, and slew many opponents during the Battle of the Blackwater. Also, Garlan is another of the warriors that gets a boost from his physical advantages; he is fairly large, as he wore Renly's armour in the battle. I suspect that Garlan is a top tier fighter, but I do not think he can be placed there before we see him fight.
2. Khal Drogo: Drogo was an elite warrior, the best of the Dothraki population, which must have numbered at least 100,000. He was a massive man who went his entire life without losing a single fight, and who survived countless battles and duels despite his culture's lack of armor or effective medicine. However, it is this culture that prevents Drogo, whose skills and physical attributes should easily place him within the top tier, from even toppin the second tier of fighters. His weaponry is primitive, his armor nonexistent, and his ability to use a variety of weapons nonexistent (due to a lack of the formal training that Westerosi nobles are granted). Even if Drogo were given Westerosi arms and armour in an attempt to level the playing field, he would be unable to utilize them effectively. On paper, Drogo is top tier material, but his equipment holds him back.
3. Gregor Clegane: The Mountain That Rides is, bar none, the strongest man in Westeros during the books, and perhaps ever. His reach is far greater than anyone else's, and his physical power makes a fight with him a dangerous prospect, even for far more skilled warriors, as one mistake means certain death. his attacks cannot be blocked by most combatants, so those that rely solely on their size and ability to take blows will fall to him easily, i.e. Victarion Greyjoy.
4. Bronn: Bronn is hard to place as a warrior. He seems highly competent, but has not faced another great fighter yet. However, his record indicates that he is an extremely capable swordsman, and this skill is augmented by his ability to properly analyze his opponents.
5. Barristan Selmy: Once tied with the Sword of the Morning in combat ability, Selmy is now an old man. His physical capabilities have deteriorated; however, his fight with the Titan's Bastard demonstrated that he remains an extraordinarily quick fighter, and that he still possesses enough strength to kill a man. His experience and combat record are, of course, second to none, and he has unsurpassed battle instincts due to the amount of combat he has seen.
6. Tie: Quorin Halfhand and Mance Rayder
Quorin Halfhand: Quorin's abilities in combat greatly surpass those of anyone else at the Wall. He is capable of improvisation during battle as well; when his fingers were severed, Quorin defeated the Wildling who took them by thrusting the bleeding stumps that remained into the opponent's eyes, blinding him. Quorin only directly demonstrated his martial skill once in the books, but it was impressive; being easily capable of defeating Jon Snow, wielding a Valyrian Steel sword, while Ghost chewed on his leg, is a noteworthy feat.
Mance Rayder: The King Beyond the Wall earned that title through feats of strength at arms. He defeated numerous other claimants to the position in battle, and was at least equal to, if not better than, Quorin Halfhand during his time on the Wall. Mance also soundly defeated Jon Snow in A Dance With Dragons, demonstrating that he was, at the time, the best warrior at the Wall, and perhaps in the entire North.
7. Oberyn Martell: The Red Viper is a difficult warrior to rank. He has a fairly lengthy record of combats, but he is only ever seen in a fight once, when he lost to the Mountain. However, Oberyn had essentially won the fight prior to his death, and only lost due to a desire to make Clegane confess his crimes. Despite Oberyn's clear superiority in his fight with the Mountain, I don't feel that he was actually remarkably better. He was certainly somewhat superior to the Mountain, but I think Gregor was the perfect opponent for Oberyn's combat style, and that Gregor would do better against many people on this list than Oberyn would. Regardless, Oberyn must be regarded as an excellent warrior, and his ranking may not reflect his actual skill level due to a lack of information.
8. Loras Tyrell: Loras may not be Garlan's equal as a swordsman, but this topic is about top tier fighters, and Loras has demonstrated a variety of combat skills. He is excellent with a variety of weapons, including the sword, morningstar, and poleaxe. He is also one of the best jousters in the realm. Despite his defeat at the hands of Brienne in the Bitterbridge tourney, I am fairly confident that Loras is better than her. If they had been using real weapons, he would have killed her several times before she tackled him and drew her dirk.
Third Tier
1. Syrio Forel
Syrio is low on my list not due to any lack of skill on his part, but due to the impracticality of his fighting style when facing armored opponents. Although there are a couple of warriors listed above Syrio that I feel he could defeat, they could in turn defeat many that he could not simply due to the extreme difficulty of fighting men in plate armour with the waterdancing techniques he employs.
2. Brienne of Tarth/Strong Belwas
Belwas has a much more extensive record, and I feel that he would slay more men in battle, but if it came to a duel Brienne would win as soon as he allowed her to strike him.
3. Balon Swann
4. Victarion Greyjoy
5. Lothor Brune
Brune may belong much higher on this list, as his feats at the Battle of the Blackwater were very impressive, but I don't think there is enough information to validate a high ranking for him yet.
6. Jon Snow
7. Jorah Mormont
8. Beric Dondarrion
9. Greatjon Umber
10. Bronze Yohn Royce
11. Iron Emmett
12. Eddard Stark
13. Jory Cassel
14. Asha Greyjoy
Not Included Due to Lack of Information/Certainty: Lyn Corbray, Areo Hotah, Andrik the Unsmiling, Black Lorrhen, Dagmer Cleftjaw, Brynden Tully, Shagga, Timmett, Gerold Dayne, Thoros of Myr, Strongboar, Harras Harlaw, Randyll Tarly.