The Greatest Warrior...

[The Might]

I'd take Eonwe, since he is considered to be the best in Arda at fighting with a weapon.
As far as only mortals are concerned...I think it's really hard to say. There are many that had proven great skill as warriors, so it's difficult for me to choose only one.
But the coolest warrior definitely has to be Aghan, who managed to defeat some Orcs, even without fighting himself.

[EluThingol]

It's a little hard to say, trying to breakdown exactly the criteria for "greatest"... you have to take into consideration the foes they were facing at the time, and how grea tthey were... how many they faced, and who their peers were at the time.

That being said, considering the fear Morgoth had about his greatness, the "legend" of his return at the end of the world, and what he was able to accomplish nearly single handedly during his many campaigns, I would say Turin.

But I would say that, according to many arguable stances, Feanor, Fingolfin, Beleg, Hurin, Ecthelion, are all valid candidates.

[Faramir Jones]

My vote for the greatest warrior goes to Húrin; because this is someone who, captured and tortured by Morgoth, refuses to beg for mercy for himself or his family; and indeed utters the most dreadful insult against him, calling him an 'escaped thrall [a serf or unfree servant, the lowest person in the social order] of the Valar'. This is against somone who calls himself King of the World.

[TheGreatElvenWarrior]

The greatest warrior is me, of course! Who else? My username comes from when Sam was called a great elven warrior, and while he wasn't really a warrior, when he had to stand up to orcs, Frodo, Gollum, whoever, he was brave and fought well.

I also think that Turin, Hurin, and Beleg would qualify as great warriors. Calling someone "the greatest warrior" is just an opinion. It is just like saying that carrots are the greatest vegetable or cookie dough is the greatest ice cream.