We use high-quality electrodynamics methods to study the extinction spectra of one-dimensional linear chains and two-dimensional planar arrays of spherical silver nanoparticles, placing emphasis on the variation of the plasmon resonance wavelength and width with array structure (spacing, symmetry), particle size, and polarization direction. Two levels of theory have been considered, coupled dipoles with fully retarded interactions and T-matrix theory that includes a converged multipole expansion on each particle. We find that the most important array effects for particles having a radius of 30 nm or smaller are captured by the couple dipole approach. Our calculations demonstrate several surprising effects that run counter to conventional wisdom in which the particle interactions are assumed to be governed by electrostatic dipolar interactions. In particular, we find that for planar arrays of particles with polarization parallel to the plane the plasmon resonance blue shifts as array spacing D decreases for D larger than about 75 nm and then it red shifts for smaller spacings. In addition, we find that the plasmon narrows for D > 180 nm but broadens for smaller spacings. The results can be rationalized using a simple analytical model, which demonstrates that the plasmon wavelength shift is determined by the real part of the retarded dipole sum while the width is determined by the imaginary part of this sum. The optimal blue shifts and narrowing are found when the particle spacing is slightly smaller than the plasmon wavelength, while red shifts and broadening can be found for spacings much smaller than the plasmon wavelength at which electrostatic interactions are dominant. We also find that the array spectrum does not change significantly with array symmetry (square or hexagonal) or irradiated spot size (i.e., constant array size or constant particle number).