Could this now business go like this:
Earth has many time zones, and yet we all experience our own 'now' moments from second to second?
It may be 15:48 right now for me, and 12:02 for you. Whether your time is before or later, can we not speak of a slice of the state of the entire earth at time t as being now?
What kind of time zones are you thinking about?

But seriously, if you are in London and it is 3pm and I am in New York and it is 10am and I drop my mug of coffee then this happens both in my "now" and your "now". That we measure the time of the day differently (time zones) has no effect on this.
However, as per special relativity, every observer must measure and map their own set of events that happen "now" just like Roboramma explained a little bit further up the thread. When two observers are in relative motion to each other, these sets will be different, and different events will be on their plane of simultaneity.
Thus, our "nows", as in the coffee mug example, align for practical purposes only. Because we move very slowly relative to each other, our planes of simultaneity are very similar and their actual misalignment is negligible. This applies for all people on Earth. In practice there is a "universal now" on Earth that all human observers can agree on.
Does this earth analogy not scale to galaxies and the universe?
It does not. The misalignment of the planes of simultaneity becomes much greater at faster relative motion. It also becomes greater at greater distances.
Imagine yours and my plane of simultaneity each as a two-dimensional plane in three-dimensional space centered on each of us. Because, on a cosmic scale, we are pretty much in the same location, so are the centres of the planes. Now consider that they are at a very, very slightly different angle. In close proximity to both of us (e.g. the Earth) the difference is so small as to be negligible. But as you move farther and farther away, the small difference in angles starts to bring the planes farther and farther apart. Even if you are only moving relative to me at a modest speed (say, you are driving by in a car), at great distances (say another galaxy) the planes are so far apart that what happens in your "now" in that galaxy could actually be a day in the future for me (for example).
The truth is that it was incredibly hard for people to let go of their preconceived notions and intuition and just follow the path laid out by experimental results.
It was indeed. For the longest time people considered the idea of time not being universal as absurd and it really is very counterintuitive. Navigator's intuition is not worse than Newton's, it's just that long and careful work has shown that our intuition does not reflect reality in this case.