We present first results from Galaxy Zoo 2, the second phase of the highly successful Galaxy Zoo project (www.galaxyzoo.org). Using a volume-limited sample of 13 665 disc galaxies (0.01 < z < 0.06 and M-r < -19.38), we study the fraction of galaxies with bars as a function of global galaxy properties like colour, luminosity and bulge prominence. Overall, 29.4 +/- 0.5 per cent of galaxies in our sample have a bar, in excellent agreement with previous visually classified samples of galaxies (although this overall fraction is lower than that measured by automated bar-finding methods). We see a clear increase in the bar fraction with redder (g - r) colours, decreased luminosity and in galaxies with more prominent bulges, to the extent that over half of the red, bulge-dominated disc galaxies in our sample possess a bar. We see evidence for a colour bimodality for our sample of disc galaxies, with a 'red sequence' that is both bulge and bar dominated, and a 'blue cloud' which has little, or no, evidence for a (classical) bulge or bar. These results are consistent with similar trends for barred galaxies seen recently both locally and at higher redshift, and with early studies using the RC3. We discuss these results in the context of internal (secular) galaxy evolution scenarios and the possible links to the formation of bars and bulges in disc galaxies.