The nature of the interaction between galaxies and the intergalactic medium (IGM) is one of the most fundamental problems in astrophysics. The accretion of gas onto galaxies provides fuel for star formation, while galactic winds transform the nearby IGM in a number of ways. One exciting technique to study this gas is through the imaging of hydrogen Lyα emission. We use cosmological simulations to study the Lyα signals expected from the growth of cosmic structure from z = 0-5. We show that if dust absorption is negligible, recombinations following the absorption of stellar ionizing photons dominate the total Lyα photon production rate. However, galaxies are also surrounded by "Lyα coronae" of diffuse IGM gas. These coronae are composed of a combination of accreting gas and material ejected from the central galaxy by winds. The Lyα emission from this phase is powered by a combination of gravitational processes and the photoionizing background. While the former dominates at z ~ 0, collisional excitation following photoheating may well dominate the total emission at higher redshifts. The central regions of these systems are dense enough to shield themselves from the metagalactic ionizing background; unfortunately, in this regime our simulations are no longer reliable. We therefore consider several scenarios for the emission from the central cores, including one in which self-shielded gas does not emit at all. We show that the combination of star formation and cooling IGM gas can explain most of the observed "Lyα blobs" at z ~ 3, with the important exception of the largest sources. On the other hand, except under the most optimistic assumptions, cooling IGM gas cannot explain the observations on its own.