observation - What uncertainty does an error bar signify in astronomy?

The most common way to represent uncertainty is with symmetric error bars around a central point. This is in turn commonly interpreted as the 95 % confidence interval. Ie, the actual data point is the centre of a Gaussian which has a width at 95 % of it's height as the size of the error bars.

This is only statistical uncertainty and often not explicitly stated. One also refers to measurement and discovery with different confidence intervals... discovery is commonly only claimed when there is a a 5-sigma confidence interval. Ie, if the measurement lies more than 5 widths of the peak away from the theory or prediction, you've made a discovery.

Note, we are leaving out here systematic uncertainty and instument bias, which can only increase the total uncertainty. Usually it is assumed that there is no correlation, so they are combined using the sum of their squares.

Long story short - always ask what the error bars represent, especially if they look too "clean".