In general, the standard error tells you how uncertain you are that true value of the top of the bar is where the bar says it is. When there are multiple bars, it can also enable comparisons between bars, in the sense of a statistical test. However, interpreting them in this way requires some assumptions, shown graphically below. If you are really interested in comparing the bars to see if the differences are statistically significant, then you should run tests on the data and display which tests were significant,like this.
In addition, I would suggest using confidence intervals rather than standard errors.
This paper is well worth the read:
Cumming and Finch. "Inference by Eye: Confidence Intervals and How to Read Pictures of Data." Am Psych. Vol. 60, No. 2, 170–180.
Their overall conclusion is: "Seek bars that relate directly to effects of interest, be sensitive to experimental design, and interpret the intervals."
For independent samples, using confidence intervals, half overlap of the CIs means the difference is statistically significant.
For independent samples using standard error bars instead, the following graph shows you how to figure out statistical significance: