Co3V2O8 (CVO) has a different type of geometrically frustrated magnetic lattice, a kagomé staircase, where the full frustration of a conventional kagomé lattice is partially relieved. The crystal structure consists of two inequivalent (magnetic) Co sites, one-dimensional chains of Co(2) spine sites, linked by Co(1) cross-tie sites. Neutron powder diffraction has been used to solve the basic magnetic and crystal structures of this system, while polarized and unpolarized single crystal diffraction measurements have been used to reveal a rich variety of incommensurate phases, interspersed with lock-in transitions to commensurate phases. CVO initially orders magnetically at 11.3K into an incommensurate, transversely polarized, spin density wave state, with wave vector k=(0,δ,0) with δ=0.55 and the spin direction along the a axis. δ is found to decrease monotonically with decreasing temperature and then locks into a commensurate antiferromagnetic structure with δ=1/2 for 6.9<8.6K. In this phase, there is a ferromagnetic layer where the spine site and cross-tie sites have ordered moments of 1.39μB and 1.17μB, respectively, and an antiferromagnetic layer where the spine-site has an ordered moment of 2.55μB, while the cross-tie sites are fully frustrated and have no observable ordered moment. Below 6.9K, the magnetic structure becomes incommensurate again, and the presence of higher-order satellite peaks indicates that the magnetic structure deviates from a simple sinusoid. δ continues to decrease with decreasing temperature and locks in again at δ=1/3 over a narrow temperature range (6.2<6.5K). The system then undergoes a strongly first-order transition to the ferromagnetic ground state (δ=0) at Tc=6.2K. The ferromagnetism partially relieves the cross-tie site frustration, with ordered moments on the spine-site and cross-tie sites of 2.73μB and 1.54μB, respectively. The spin direction for all spins is along the a axis (Ising-like behavior). A dielectric anomaly is observed around the ferromagnetic transition temperature of 6.2K, demonstrating that there is significant spin-charge coupling present in CVO. A theory based on group theory analysis and a minimal Ising model with competing exchange interactions can explain the basic features of the magnetic ordering.