Differential measurements of charged particle azimuthal anisotropy are presented for lead-lead collisions at root sNN = 2.76 TeV with the ATLAS detector at the LHC, based on an integrated luminosity of approximately 8 mu b(-1). This anisotropy is characterized via a Fourier expansion of the distribution of charged particles in azimuthal angle relative to the reaction plane, with the coefficients v(n) denoting the magnitude of the anisotropy. Significant v(2)-v(6) values are obtained as a function of transverse momentum (0.5 < p(T) < 20 GeV), pseudorapidity (|eta| < 2.5), and centrality using an event plane method. The v(n) values for n >= 3 are found to vary weakly with both eta and centrality, and their p(T) dependencies are found to follow an approximate scaling relation, v(n)(1/n)(p(T)) proportional to v(2)(1/2)(p(T)), except in the top 5% most central collisions. A Fourier analysis of the charged particle pair distribution in relative azimuthal angle (Delta phi = phi(a)-phi(b)) is performed to extract the coefficients v(n,n) = < cos n Delta phi >. For pairs of charged particles with a large pseudorapidity gap (|Delta eta = eta(a) - eta(b)| > 2) and one particle with p(T) < 3 GeV, the v(2,2)-v(6,6) values are found to factorize as v(n,n)(p(T)(a), p(T)(b)) approximate to v(n) (p(T)(a))v(n)(p(T)(b)) in central and midcentral events. Such factorization suggests that these values of v(2,2)-v(6,6) are primarily attributable to the response of the created matter to the fluctuations in the geometry of the initial state. A detailed study shows that the v(1,1)(p(T)(a), p(T)(b)) data are consistent with the combined contributions from a rapidity-even v(1) and global momentum conservation. A two-component fit is used to extract the v(1) contribution. The extracted v(1) isobserved to cross zero at pT approximate to 1.0 GeV, reaches a maximum at 4-5 GeV with a value comparable to that for v(3), and decreases at higher p(T).