Books about the nature of time fall into two camps - those that argue for or assume the reality of time, and those that argue that time is not part of the fundamental fabric of the universe.

In the former camp are:

Time's Arrow and Archimedes' Point - Huw Price - an accessible explanation of how a direction of time emerges in cosmology and quantum physics, which implicitly assumes the reality of time.

From Eternity to Here - Sean Carroll - argues for a cyclical "multiverse" in which the apparent direction of time is determined by the direction in which entropy is increasing.

The Singular Universe and the Reality of Time - Roberto Mangabeira and Lee Smolin - an argument for the reality of time and its key role in a theory of quantum gravity that also strongly opposes the idea of a "multiverse".

In the latter camp are:

The Order of Time - Carlo Rovelli - short and accessible with little or no mathematics, Rovelli argues that time is an emergent phenomenon that arises from how we perceive the universe.

The End of Time - Julian Barbour - an argument for a timeless block universe in which time and history arise from the second law of thermodynamics.

And finally a book that has a foot in both camps:

About Time - Paul Davies (mentioned in another answer) - a survey of the role of time in relativity, cosmology and quantum physics. In the argument about the reality of time, Davies sits on the fence.