Recently I was engaged in a discussion on the meaning of character level in RPGs...it was a lively discussion, but rather different from the original topic of the thread. So as to avoid further derail, I decided to create a thread to discuss it in its own right.
I am a veteran of the early editions of D&D, and like my campaigns rooted in gritty realism. I eschew the character-as-superhero ethos. In my estimation, a first-level character is no hero, but rather a raw recruit who barely knows which end of the weapon to hold. He really isn't that much better than the no-level/0-level populace of the realm. He has an edge in ability scores (whilst the mean ability score of the average human is 10 to 11, most adventuring parties I DM probably average closer to 13. A cut above, but certainly no 16 to 19 across the board superhero.
In my worlds, it isn't until a character reaches third or even fourth level that she can be rightly called a skilled veteran of adventuring--a journeyman, if you will. By then she is usually able to specialize in a weapon, and has been given one or two traits (bonuses I grant to characters based on their adventuring style and action). Even then, she is quite mortal; there is no leaping tall buildings in a single bound. She begins to gain a reputation, if only locally. A rogue at this level is a good cutpurse. A warrior might be made a squire, or the culture's equivalent.
By fifth to sixth level, a character begins to make a footprint on the world, if a modest one. He is talked about in taverns; perhaps his name is dropped in passing at the courts of some minor nobles. By then, the character has certainly specialized, if not double-specialized in a weapon, and has developed further traits. A rogue at this level can cause no small amount of mischief. A priest at this level begins to gain a following, and most likely a low office in the clerical hierarchy. A wizard/MU at this level now has access to some increasingly potent magic, and can do things that get talked about. For warriors, a knighthood is a distinct possibility by now. However, the character is still quite mortal.
In modern editions of D&D, a 6th-level character is a superhero in all but name. This is why I prefer earlier editions.