As other have mentioned, F_1 does not exist of a field. Tits conjectured the existence of a "field of characteristic one" F_1 for which one would have the equality G(F_1) = W, where G is any Chevalley group scheme and W its corresponding Weyl group.
Later on Manin suggested that the "absolute point" proposed in Deninger's program to prove the Riemann Hypothesis might be thought of as "Spec F_1", thus stating the problem of developing an algebraic geometry (and eventually a theory of motives) over it.
There are several non-equivalent approaches to F_1 geometry, but a common punchline might be "doing F_1 geometry is finding out the least possible amount of information about an object that still allows to speak about its geometrical properties". A "folkloric" introduction can be found in the paper by Cohn Projective geometry over F_1 and the Gaussian binomial coefficients.
It seems that all approaches so far contain a common intersection, consisting on toric varieties which are equivalent to schemes modeled after monoids. In the case of a toric variety, the "descent data" that gives you the F_1 geometry is the fan structure, that can be reinterpreted as a diagram of monoids (cf. some works by Kato). What else are F_1 varieties beyond toric is something that depends a lot on your approach, ranging from Kato-Deitmar (for which toric is all there is) to Durov and Haran's categorical constructions which contains very large families of examples. A somehow in-the-middle viewpoint is Soule's (and its refinement by Connes-Consani) which in the finite type case is not restricted only to toric varieties but to something slightly more general (varieties that can be chopped in pieces that are split tori). No approach is yet conclusive, so the definitions and families of examples are likely to change as the theory develops.
Last month Oliver Lorscheid and myself presented an state-of-the-art overview of most of the different approaches to F_1 geometry: Mapping F_1-land: An overview of geometries over the field with one element (sorry for the self promotion).
There have been several questions on mathoverflow about the field with one element. Of course, such a field doesn't really exist and the discussion must fray sooner or later. So here is a different kind of answer.
Besides finite fields, which are 0-manifolds, there are only two fields which are manifolds, $\mathbb{C}$ and $\mathbb{R}$. There is a generalization of cardinality for manifolds and similar spaces, namely the geometric Euler characteristic. (This is as opposed homotopy-theoretic Euler characteristic; they are equal for compact spaces.) The geometric Euler characteristic of $\mathbb{C}$ is 1, while the geometric Euler characteristic of $\mathbb{R}$ is -1. In this sense, $\mathbb{C} = \mathbb{F}_1$ while $\mathbb{R} = \mathbb{F}_{-1}$.
It works well for some of the motivating examples of the fictitious field with one element. For instance, the Euler characteristic of the Grassmannian $\text{Gr}(k,n)$ over $\mathbb{F}_q$ is then uniformly the Gaussian binomial coefficient $\binom{n}{k}_q$.
In this interpretation, $\mathbb{F}_1$ is algebraically closed. It is also a quadratic extension of $\mathbb{F}_{-1}$; the generalized cardinality squares, as it should.
Best Answer
If you remove the overline, you have the affine scheme Spec Z. It is the spectrum of a noetherian domain of Krull dimension one. This description also holds for any affine algebraic curve over a field, so we have the basis for an analogy. Z has many structural features in common with the ring of polynomials with coefficients in a field (e.g., basic number-theoretic machinery like a Euclidean algorithm), so there are good reasons to think of Spec Z as a curve.
However, there are several differences. First, Z is the initial object in the category of commutative rings, so Spec Z is the final object in the category of locally ringed spaces. In particular, Spec Z is not a curve over any base field. Second, there are maps from the spectra of fields of many different characteristics into Spec Z. This makes objects like zeta functions and L-functions much more transcendental (in appearance), since the logarithm doesn't behave as nicely as it does over finite fields. You can also find a definition of genus of a number field in Neukirch's Algebraic Number Theory, and while it is zero for Q, it tends to be transcendental in general (essentially due to the presence of logs).
Despite the visible flaws in the analogy, there are good reasons for thinking that Spec Z should have a compactification, in a manner similar to the compactification of Spec F[t] into the projective line. The most basic is that Z has a valuation that is not captured by the points of the topological space, namely the usual Archimedean absolute value. For the ring of polynomials in a field, this translates to the degree. By adding this extra valuation you find that if you take all of the absolute values of a rational number (normalized appropriately) and take their product, you get one. Logarithmically, the sum of valuations is zero just like the residue theorem in the function field case. A more sophisticated reason comes from arithmetic intersection theory. If you take a curve defined over Z, you can compute a notion of intersection between two integral points, in an manner analogous to computing the intersection class of two curves in a surface. In the topological setting, you need some conditions for this to be well-behaved, e.g., the surface should be compact, so you can't push the intersections off the end of the surface. This is achieved over Z using Arakelov theory - the intersection over infinity is computed by base changing the curve to the complex numbers, making some distribution on the resulting Riemann surface using the integral points, and computing the integral of that distribution.
I don't think there is general agreement on how to make a geometric object with all of the properties we want from a compactification of Spec Z. In particular, there doesn't seem to be a satisfactory theory of the "base field" yet. There is a way to approximate the compactification using Berkovich spaces, which are an analytic refinement of schemes. The Berkovich spectrum of Z already comes with an Archimedean branch, and if you weaken the triangle inequality, you can have points corresponding to arbitrary non-negative powers of the Archimedean absolute value. Compactification means adding the point $|-|^\infty_\infty$. The local ring at this point is the closed interval [-1,1] in R, with the multiplication operation (i.e., you have some kind of logarithmic structure, instead of an actual ring), and the global functions on this object are given by the multiplicative monoid {-1,0,1}, which is a rather boring looking candidate for the field with one element.