We examine intergalactic clouds confined by the gravity of cold dark matter (CDM), the so-called minihalo, as a unified model for neutral hydrogen (H I) absorption systems of quasars. The reported H I column density distribution from 10(13) cm-2 of the Lyalpha forest to 10(22) cm-2 of damped Lyalpha systems is well reproduced by the evolution of the two-phase nature of minihalos: a neutral core surrounded by an ionized envelope. Shielding effects of diffuse UV flux due to neutral hydrogen in the envelope cause the appearance of a neutral core in a massive minihalo, and this core is detected as a high column density system. Highly ionized envelopes and/or less massive one-phase minihalos correspond to the Lyalpha forest. In order to reproduce the number density evolution of the Lyalpha forest and the redshift variation of the continuum depression of high-redshift quasars, minihalos must experience an expansion that is caused by the increase of UV flux with the decrease of redshift at z > 2. Assuming a simplified evolution law of UV flux and mass function of gas clouds, we can reproduce all these observational properties by the minihalo model. At z < 2, the diffuse UV flux decreases and minihalos are apt to contract. The collapse of neutral cores is delayed, especially for small mass clouds, as late as z less than or similar to 1, and these are considered to become dwarf galaxies. In this picture we present some predictions about low z Lyalpha clouds for observations of the Hubble Space Telescope.