Is the meter relative when we are near the speed of light?
I was reading a physics book and I found that the meter is the length that light travels for an amount of time, so since time is relative near the speed of light, does it mean that the meter will be inconsistent at different speeds?
So for example, a meter measured near a black hole will be longer to a meter measured on the Earth?
It depends.
Suppose you are carrying a meter stick with you. When you are at rest with respect to this meter stick, its size is always the same. It doesn't matter if you're near a black hole, near the speed of light, or anything of the sort. (I'm, of course, ignoring the effects on the material and considering an ideal meter stick).
Nevertheless, if you leave the meter stick on the ground and run next to it along the direction of the meter stick, the meter stick will be shorter (it won't "appear" shorter, it will be shorter). When something has motion relative to you, this something experiences a phenomenon known as length contraction, which means it is contracted by relativistic effects. The contraction is more intense at larger speeds and effectively imperceptible at low speeds relative to the speed of light (which is why you never noticed it).
Hence, the size of a meter stick depends on your motion relative to it. However, if you are sufficiently close to it and at rest relative to it, a meter is a meter, anywhere in the Universe (to the best of our knowledge).
I should mention that the reason behind this is the fact that the speed of light does not depend on the reference frame. The speed of light is the same, no matter how fast you are moving relative to something else. Length contraction is a consequence of this constancy.
The short answer is that a meter is always a meter by definition. The modern definition of the meter in terms of the speed of light doesn't change that.
A longer answer:
One of the lessons from relativity is that you cannot compare two frames moving with respect to each other by looking only at the spatial coordinates. Length is about the difference between two spatial coordinates at a fixed time.
If you have a meter stick that you carry with you, you are in its rest frame and it will appear to be 1 meter. If you go speeding past someone else who has their own meter stick and try to compare the lengths, then you'll essentially have a version of the "pole paradox", which is resolved by a combination of how each observer measures length and whether they agree on the time at which the length measurement was made.
I was reading a physics book and I found that the meter is the length that light travels for an amount of time, so since time is relative near the speed of light, does it mean that the meter will be inconsistent at different speeds?
Time is relative, but so is velocity. If to you the meter rod is travelling near the speed of light, then to an observer comoving with the rod, the rod is stationary. If you carry a (temperature corrected) meter rod with you, then whatever velocity you appear to have or whatever gravitational field you are in, it will still measure a metre. To other observers with a different velocity to you or at a different height in a gravitational field, your rod can appear shorter.
So how do you check the length of your personal meter rod? You can place a mirror at one end and send a single from the other end and time how long the signal takes to go to the mirror and back. This will always be the same time interval$^*$. Other observers will say you have not noticed the length contraction of your rod, because your clock is time dilated and your notion of simultaneity is different to theirs.
$^*$ There is an exception to this rule. If you are in an extremely strong gravitational field like near a black hole, the speed of light is only constant in your local vicinity. The stronger the field, the smaller this local area and in the extreme case this local area might be less than a meter.
