The complicated sensing and communication environment of power systems results in measurement errors with unknown, nonzero-mean, non-Gaussian, and time-varying statistics. Traditional state estimator designs are based on heuristic assumptions of measurement error distributions and are agnostic to the true error statistics, yielding suboptimal error filtering performances in reality. This article investigates the supervisory control and data acquisition (SCADA) and phasor measurement unit (PMU) measurement chain modeling and presents a new state estimation (SE) paradigm based on the concept of adaptive SE (ASE). Instead of ignoring or passively resisting the unknown measurement error statistics, it proactively captures this information and adapts the structure and parameters of the estimator online to optimize the accuracy of the state estimates. The proposed method can capture arbitrarily complex measurement error distributions, preserves high computational efficiency, adapts to abrupt gross errors, and also enables a sensor calibration approach for both PMUs and SCADA without the need for field experiments. The proposed method is validated on the IEEE 30-bus test system with complex and time-varying measurement errors generated by comprehensive SCADA and PMU measurement chain modeling.