The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. Aesthetically, this seems kind of ugly to me. The real line is a high-tech piece of mathematical machinery. We build up all that structure, then build the definition of a manifold out of it, then throw away most of the structure. It seems kind of like building an airplane by taking a tank, adding wings, and getting rid of the armor and the gun turret.
I've spent some time trying to figure out a definition that would better suit my delicate sensibilities, and have come up with the following sketch: An $n$-dimensional manifold is a completely normal, second-countable, locally connected topological space that has Lebesgue covering dimension $n$, is a homogeneous space under its own homeomorphism group, and is a complete uniform space.
Does this work? I should reveal at this point that I'm a physicist, and no more than a pathetic dilettante at math. I have never had a formal course in topology. My check on my proposed definition consisted of buying a copy of Steen's Counterexamples in Topology and flipping through it to try to find examples that would invalidate my proposed definition.
Since I'm not competent as a mathematician, what would probably be the best outcome of this question would be if someone could point me to a book or paper in which my idea is carried out by someone competent.
Clarification: I mean a topological manifold, not a smooth manifold.
Also, I should have mentioned in my original post that I had located some literature on the $n=1$ case in terms of characterizing the real line (which is not, of course, the same as characterizing a 1-dimensional manifold, but is a related idea):
P.M. Rice, "A topological characterization of the real numbers," 1969
S.P. Franklin and G.V. Krishnarao, "On the Topological Characterization of the Real Line: An Addendum," J. London Math Soc (2) 3 (1971) 392.
Brouwer, "On the topological characterization of the real line," http://repository.cwi.nl/search/fullrecord.php?publnr=7215
Kleiber, "A topological characterization of the real numbers," J. London Math Soc, (2) 7 (1973) 199