Maximise the log-likelihood, since independence makes the likelihood a product and logs turn it into a differentiable sum.
The classic interview fact. For normal data, the MLE of the mean is the sample mean (unbiased), but the MLE of the variance divides by n, not n-1, and is therefore biased downward. Bessel's correction restores unbiasedness at the cost of no longer being the MLE - the two criteria genuinely conflict.
Properties: consistent, asymptotically efficient, and invariant under reparameterisation. Not necessarily unbiased in finite samples.
The failure mode: model misspecification gives a confidently wrong answer with wrong confidence intervals.