(S)-glycidyl benzyl ether NMR problem
The pattern is neither a true quartet nor a doublet of doublets: it is two doublets. Many thanks to long for pointing out that it is also frequently described as an "AB quartet".

The peaks arise due to the benzylic protons, i.e. the $\ce{CH2}$ between the ether oxygen and the phenyl group. Due to the existing stereocentre on the epoxide, these two protons are rendered diastereotopic: due to their different position relative to the stereocentre, they experience a different net magnetic field, and hence have a different chemical shift. Therefore, in the absence of coupling, one expects to see two singlets.

If we add in spin-spin coupling, then we see that both protons are capable of coupling to each other. The magnitude of the geminal coupling $^2J$ is generally somewhere between 10 to 15 Hz. In this case the zoomed-in diagram shows that the coupling is roughly 12 Hz. So, we now expect to see two doublets.

Why, then, are the intensities of the two peaks in each doublet different? This is due to so-called "roofing": when the difference in the resonance frequencies of two protons is comparable to the coupling constant between them, the shape of the multiplets will be such that they have a "slant" towards each other. In this case, the difference in resonance frequencies is roughly 20 Hz (measure the distance between the centre of each doublet), which is on the same order of magnitude as the coupling constant.

This can, to the best of my knowledge, only be explained with quantum mechanics: see What is the origin of &#39;tenting&#39; or &#39;roofing&#39; in NMR spectra? (the question has more examples).