The easiest to understand diagnostic used to measure publication bias is the funnel plot. Suppose the true effect of interest is theta = 0.2. Then the observed effects in studies should be centered at 0.2; some will be higher, some will be lower. Assuming no systematic error, the degree to which study results vary around 0.2 should be proportional to the precision of the study (think sampling error given a sampling design). A hypothetical study of an infinitely large meta-population would produce the effect estimate of exactly 0.2, infinitely precisely. A series of very small studies will likely show quite divergent results, just on the basis of precision.
A funnel plot plots effect sizes on the x axis and precision on the y axis. The most precise studies should be tightly grouped around the meta-analytic average effect; the least precise studies should be spread more widely. This forms a triangular, funnel shape. If no publication bias exists, the spread of studies below the magnitude of the average effect should be comparable to the spread of studies above the magnitude of the average effect.
If there is publication bias, then the points that would form the left (without loss of generality; right if negative effect size) portion of the funnel will not be observable.
There are issues with funnel plots and there are other diagnostics but I hope this provides insight into one of the tools used. Notably, as a diagnostic, funnel plots work whether the true effect is positive, negative, or null; they assume only that the underlying assumptions of meta-analysis are true (that studies represent a sample of the same, true underlying effect -- other diagnostics and corrections exist when this is violated)