We present a technique to identify optical counterparts of 250-mu m-selected sources from the Herschel-ATLAS survey. Of the 6621 250 mu m > 32-mJy sources in our science demonstration catalogue we find that similar to 60 per cent have counterparts brighter than r = 22.4 mag in the Sloan Digital Sky Survey. Applying a likelihood ratio technique we are able to identify 2423 of the counterparts with a reliability R > 0.8. This is approximately 37 per cent of the full 250-mu m catalogue. We have estimated photometric redshifts for each of these 2423 reliable counterparts, while 1099 also have spectroscopic redshifts collated from several different sources, including the GAMA survey. We estimate the completeness of identifying counterparts as a function of redshift, and present evidence that 250-mu m-selected Herschel-ATLAS galaxies have a bimodal redshift distribution. Those with reliable optical identifications have a redshift distribution peaking at z approximate to 0.25 +/- 0.05, while submillimetre colours suggest that a significant fraction with no counterpart above the r-band limit have z > 1. We also suggest a method for selecting populations of strongly lensed high-redshift galaxies. Our identifications are matched to UV-NIR photometry from the GAMA survey, and these data are available as part of the Herschel-ATLAS public data release.