yakin said:
Does gravity on Earth stays constant regardless of height?. For instance, if a person standing at a mountain would he experiences the same gravity as the person standing on the ground?
Not quite, but almost. For instance, astronauts on the ISS still feel gravity (otherwise they would be flung into outer space) but at only about $8.76 \mathrm{m}/\mathrm{s}^2$, compared to the $9.81 \mathrm{m}/\mathrm{s}^2$ surface gravity. The reason they are weightless is because the rest of the station is also moving with them, so they have (near) zero acceleration in the station's reference frame, and the reason they don't crash into the Earth is because they are in orbit.
The formula to calculate gravity at an arbitrary height $h > 0$ can be obtained from the Earth's surface gravity ($g_0 = 9.81 \mathrm{m}/\mathrm{s}^2$) and its radius $r = 6371 \mathrm{km}$, and is as follows:
$$g_h = g_0 \left ( \frac{r}{r + h} \right )^2$$
This can be obtained from Newton's law of gravitation, basically since gravity is inversely proportional to distance squared, you can take the squared ratio of the distances to get a ratio of gravity at different heights. This is assuming the Earth is a perfect sphere, which is not quite true, but anyway, plugging in some numbers, say $h = 9 \mathrm{km}$ (Mount Everest):
$$g_{9 \mathrm{km}} = 9.78$$
So, even standing on the highest mountain, the Earth's gravity is still very strong.
However, because the Earth is actually not a perfect sphere, but is elongated at the equator, people living at the poles weigh more than people on the equator (by a small but measurable amount). As a less extreme example, take your average person who weighs $60 \mathrm{kg}$ and lives in, say, southern Australia, where gravity is approximately $9.8 \mathrm{m}/\mathrm{s}^2$. If he were to travel to central Africa (the equator), he would feel slightly lighter, as if he only weighed $59.88 \mathrm{kg}$. Of course his mass didn't change, but gravity has, and so his scale (unless calibrated to the new gravity somehow) will show that he weighs a couple hundred grams less :D
Of course this isn't quite true because there's more than gravity at work here, there are also centrifugal effects from the Earth's rotation, and the fact that gravity does not depend strictly on distance from the Earth's center (it also depends on the mass distribution of the Earth, which is actually quite complicated), etc... but to answer your question, yes, gravity does get weaker as you go further away from the Earth, but it isn't noticeable unless you go a few hundred kilometres high.