I am studying different point estimate methods and read that when using MAP vs ML estimates, when we use a "uniform prior", the estimates are identical. Can somebody explain what a "uniform" prior is and give some (simple) examples of when the MAP and ML estimators would be the same?
It is a uniform distribution (either continuous or discrete).
If you use a uniform prior on a set that contains the MLE, then MAP=MLE always. The reason for this is that under this prior structure, the posterior distribution and the likelihood are proportional.
MLE is the estimate of occurrence of given event given a parameter, whereas MAP is estimate of a parameter given an event. When we use Bayes theorem further while estimating MAP it boils down to $P(D|\theta)P(\theta)$ where $P(\theta)$ is the only additional term with respect to MLE. The mean and variance estimate of MAP will be same as mean and variance estimate of MLE as Prior is remaining the same every time and is not changing at all. Thus it only acts as a constant and thus plays no role in affecting the value of mean and variance.