A meta-regression looks at a study level predictor of outcome and sees if the effect size varies according to that study criterion.
E.g. you're looking at the effects of stimulants on ADHD. You find that the pooled effect shows that stimulants seem to be effective. Does it matter which stimulant? You do a meta-regression, which lets you compare Adderall, Ritalin, etc.
A sensitivity analysis is like a sensitivity analysis anywhere. 8 of my ADHD studies were done in the US, 2 were done outside the US. If I exclude the studies that were done outside the US, does that alter my conclusions?