If each value represents a different individual, you probably want to show the variation among values. Even if each value represents a different lab experiment, it often makes sense to show the variation.
With fewer than 100 or so values, create a scatter plot that shows every value. What better way to show the variation among values than to show every value? If your data set has more than 100 or so values, a scatter plot becomes messy. Alternatives are to show a box-and-whiskers plot, a frequency distribution (histogram), or a cumulative frequency distribution.
What about plotting mean and SD? The SD does quantify variability, so this is indeed one way to graph variability. But a SD is only one value, so is a pretty limited way to show variation. A graph showing mean and SD error bar is less informative than any of the other alternatives, but takes no less space and is no easier to interpret. I see no advantage to plotting a mean and SD rather than a column scatter graph, box-and-wiskers plot, or a frequency distribution.
Of course, if you do decide to show SD error bars, be sure to say so in the figure legend so no one will think it is a SEM.
If your goal is to compare means with a t test or ANOVA, or to show how closely our data come to the predictions of a model, you may be more interested in showing how precisely the data define the mean than in showing the variability. In this case, the best approach is to plot the 95% confidence interval of the mean (or perhaps a 90% or 99% confidence interval).
What about the standard error of the mean (SEM)? Graphing the mean with an SEM error bars is a commonly used method to show how well you know the mean, The only advantage of SEM error bars are that they are shorter, but SEM error bars are harder to interpret than a confidence interval.
Whatever error bars you choose to show, be sure to state your choice. Noticing whether or not the error bars overlap tells you less than you might guess.