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Magnetic braking is essential for cloud contraction and star formation. Ambipolar diffusion is unavoidable in self-gravitating, magnetic clouds and leads to single-stage (as opposed to hierarchical) fragmentation (or core formation) and protostar formation. Magnetic forces dominate thermal-pressure and centrifugal forces over scales comparable to molecular cloud radii. Magnetic support of molecular clouds and the imperfect collisional coupling between charged and neutral particles introduce a critical magnetic length scale (λ<SUB>M,cr</SUB> = 0.62υ<SUB>A</SUB>τ<SUB>ff</SUB>) and an Alfvén length scale ((λ<SUB>A</SUB> = πυ<SUB>A</SUB>τ<SUB>ni</SUB>), respectively, in the problem which together with a critical thermal length scale (λ<SUB>T,cr</SUB> = 1.09C<SUB>a</SUB>τ<SUB>ff</SUB>) explain naturally the formation of fragments (or cores) in otherwise quiescent clouds and determine the sizes and masses of these fragments during the subsequent stages of contraction. (The quantity υ<SUB>A</SUB> is the Alfvén speed, τ<SUB>ni</SUB> the mean neutral-ion collision time, C<SUB>a</SUB> the adiabatic speed of sound, and τ<SUB>ff</SUB> the free4all time scale.) Numerical calculations based on new adaptive-grid techniques follow the formation of fragments by ambipolar diffusion and their subsequent collapse up to an enhancement in central density above its initial equilibrium value by a factor ≃10<SUP>6</SUP> with excellent spatial resolution. The results confirm the existence and relevance of the three length scales and extend the analytical understanding of fragmentation and star formation derived from them. The ultimately bimodal opposition to gravity (by magnetic forces in the envelope and by thermal-pressure forces in the core) introduces a break in the slope of the log p<SUB>n</SUB> -log r profile. The relation B<SUB>c</SUB> ∞ p<SUP>k</SUP><SUB>c</SUB> between the magnetic field strength and the gas density in cloud cores holds with K = 0.4 - 0.5 even in the presence of ambipolar diffusion up to densities ∼10<SUP>9</SUP> cm<SUP>-3</SUP> for a wide variety of clouds. The value K ≃ ½ is fairly typical. At the late stages of evolution, for example, at a central density of about 3 × 10<SUP>8</SUP> cm<SUP>-3</SUP>, a typical core is relatively uniform, contains 0.1 M<SUB>sun</SUB> and a magnetic field ≃ 3 mG, and is surrounded by a spatially rapidly decreasing, highly nonspherical (disklike) density distribution. The amount of mass available for accretion onto the compact core is limited by magnetic forces, and is typically ≃1 M<SUB>sun</SUB>. These results are built into the detailed scenario for star formation described recently elsewhere.