The effects of the dust on the determination of elemental abundances in damped Ly alpha (DLA) absorbers are investigated. Relations between the observed abundances measured in the gas phase and the overall abundances (gas plus dust) are derived as a function of dust-to-gas ratio, metallicity, element-to-element abundance pattern, average extinction coefficient of dust grains, and chemical composition of dust grains. A method is presented for determining dust-to-gas ratios, dust-to-metals ratios, and dust-corrected relative abundances in DLA absorbers by assuming dust of Galactic type and constant abundance ratios between iron-peak elements. The method is applied to a sample of 17 DLA absorbers with available Zn, Cr, and/or Fe measurements. The resulting dust-to-gas ratios are mostly distributed between 2% and 25% of the Galactic value, in good quantitative agreement with the results from reddening studies of QSOs with foreground DLA absorption. A correlation is found between dust-to-gas ratio and metallicity in DLA galaxies, with a typical dust-to-metals ratio of approximate to 60% of the Galactic value. The derived dust-to-metals ratios are then used to correct from the effects of dust the abundance ratios [Si/Fe], [S/Fe], [Ti/Fe], [Mn/Fe], and [Ni/Fe] available for a subsample of nine absorbers. The [alpha/Fe] ratios corrected from dust do not show the enhancement characteristic of metal-poor Galactic stars, but instead have essentially solar values, within +/-0.2 dex. This suggests that the chemical history of DLA absorbers is different from that experienced by the Milky Way. Evidence that points to dwarf galaxies, rather than to spiral galaxies, as important contributors to the DLA phenomenon is summarized.