We study analytically defect polariton states in Bragg multiple quantum well structures and defect-induced changes in transmission and reflection spectra. Defect layers can differ from the host layers in three ways: exciton-light coupling strength, exciton resonance frequency, and interwell spacing. We show that a single defect leads to two local polariton modes in the photonic band gap. These modes cause peculiarities in reflection and transmission spectra. Each type of defect can be reproduced experimentally, and we show that each of these plays a distinct role in the optical properties of the system. For some defects, we predict a narrow transmission window in the forbidden gap at the frequency set by parameters of the defect. We obtain analytical expressions for corresponding local frequencies as well as for reflection and transmission coefficients. We show that the presence of the defects leads to resonant tunneling of the electromagnetic waves via local polariton modes accompanied by resonant enhancement of the field inside the sample, even when a realistic absorption is taken into account. On the basis of the results obtained, we make recommendations regarding the experimental observation of the effects studied in readily available samples.