Meaning of "time existing locally relative to a reference frame" I'm reading a book in which it's written

Relativity shows that time exists locally, relative to a given reference frame.

What does this mean? I'm a bit confused because this seems like a conclusion that's presented before the actual explanation. Is there a technical term for this, so that I can look up more about it on the internet? As of now it seems by "time existing locally" the author's referring to proper time (even though I don't know what that is, exactly) through a cursory reading of the proper time wiki page.
 A: I think your book is referring to what is more formally known as the relativity of simultaneity.
In special and general relativity, time (which is a dimension of space-time) always exists. The difference from Newtonian (non-relativistic) physics is that in relativity not all observers agree on how to define the time co-ordinate i.e. how to "slice" space-time along surfaces where the time-co-ordinate is constant.
Events in space-time with the same time co-ordinate are said to be "simultaneous". In Newtonian physics a pair of events in space-time is either simultaneous or not, independent of the motion of an observer. However, in relativistic physics the attribute of simultaneity depends on the observer's motion, so simultaneity is relative.
In special relativity (i.e. without taking into account gravitational fields) two observers who are stationary relative to one another will agree on a time co-ordinate, but two observers who are moving relative to one another will not agree, as each is measuring time relative to a different inertial frame.
In general relativity (taking into account gravitational fields) then the situation is complicated by the curvature of space-time. This is probably why your book says that time only "exists" locally i.e. only observers who are stationary relative to one another and who are sufficiently close to one another will agree on whether two events in space-time are or are not simultaneous.
A: The phrase 'time exists locally' is poorly though-out and confusing as a consequence. Time exists everywhere. The local nature of time refers to the relativity of simultaneity. Where two reference frames are moving with respect to each other, clocks synchronised in one frame will in general disagree with clocks synchronised in the other. The discrepancy between the two sets of clocks varies with position, so in that sense the relative time between the two frames is local. 
A: From a Google search, it appears that your "quote" [which I have bolded] is the second part of a sentence (the italics is original):

from "Core Principles of Special and General Relativity" (James H. Luscombe) ISBN:1138542946
  
Relativity shows there is no absolute meaning to the "same time". Absolute time does not exist--it's not true that time exists independently of anything else. Time is not a parameter provided by the universe, as it is in pre-relativistic physics; relativity shows that time exists locally, relative to a given reference frame.screenshot below from page 23(?)


As the earlier answers have deduced, the author is talking about how, according to [Einsteinian] Relativity absolute simultaneity [a global notion of simultaneity] does not physically exist. That is, time is not dictated universally by the ticking of a giant clock [like the one at Big Ben], but rather it's dictated locally, personally, by a wristwatch worn by an observer. (As Bondi says in Relativity and Common Sense, "we must be prepared for universal public time to break up into a multitude of private times".) Minkowski used Eigenzeit, that is "proper time" or "one's own time"... with "proper" referring to property or ownership (not as in proper vs improper, as in correctness). 
From looking at the table of contents on Google,...
this passage is in "1.3 Space, Time, and Spacetime (p.23?)" in "Ch 1. Relativity: A theory of space, time, and gravity". The text is first giving you the big picture conclusion, then proceeds to develop an explanation for it by developing spacetime diagrams (in 1.4...) and special relativity (Ch 2).
In my opinion, this is a reasonable development of ideas,
which may be more clear when this passage is viewed in its context.
(After transcribing parts and taking a screenshot and writing an answer,
I just saw the comment by @JdeBP pointing to this book by Luscombe.)
