If there was to be a habitable planet, similar to earth, but with varying gravity at different points on its surface - lets pretend that it varies from .8g to 1.2g - what sort of terrain or form would be necessary in order for this to exist? it would still need to have a large amount of liquid water, so mechanically a very tall mountain range would need to not fall under it's own weight (and ideally also have a breathable atmosphere at such heights). It could spin much faster on it's axis, and have some sort of Centrifugal force, which would reduce the gravity near the equator, but also mess up any sort of day/night cycle, and again, potentially the atmosphere. Are there any alternatives? some sort of weird but feasible planet geometry? a large moon that can reduce the gravity on the surface of the planet depending on it's position in orbit?
TL/DR: spinning faster is probably the best way. (And you don't need that much, anyway.)
The reason most actual planets are spherical (or nearly spherical, when there are outside forces perturbing the inner gravity - as there will be in most of the examples below) is because such a huge pressure means that they almost behave as a drop of liquid. This is called hydrostatic equilibrium, and is pretty much required for a planet to have a gravity as high as 0.8 g in the first place.
If your planet is non-spherical, such as Jinx (which seems to be what you're looking for), surface gravity can vary a lot; but it is debatable whether such a planet will be able to stay in such a shape without going back to near-sphericity. (Rocks might not be rigid enough.)
If you're prepared to deal with a little handwaving, just copy the Jinx situation. Niven tried to keep the underlying physics as realistic as possible, so it's really only the "rocks might not be rigid enough" part that is a problem. (And it will be a problem in any similar case of "weird planet geometry". At least Jinx definitely falls under the "feasible" part.)
However, it does appear that rocks are not rigid enough (and there's another problem, which I'll explain in the next section); in which case we're forced to consider other possibilities - some of which you have already mentioned...
Weird planet geometry
For starters (this doesn't really fit into the previous section), a bit on weird planet geometry.
As mentioned above, a Jinx-shaped (or otherwise weirdly shaped) planet will not be in hydrostatic equilibrium (for its current position). In other words, it has to be rigid enough not to behave like a drop of liquid.
However, its atmosphere (if it has one) will not be rigid, and will basically behave as a fluid. In other word, it will flow down to the "lower" parts (those with the higher gravity), and will basically disappear from the places with the lower gravity.
So your weirdly-shaped world will (if it's large enough) have thick atmosphere in the high-gravity parts, and near-vacuum in the low-gravity parts... just like Jinx does, actually. (Or like your mountain range example, which is actually pretty much talking about the same thing - just slightly differently.)
The obvious way to have differing gravity is, yes, rapid rotation. You don't actually need that rapid to make a big difference (especially since the relevant force is proportional to the square of the angular velocity), especially if your planet has a low enough density (for now, I'll snip the formulas that explain why I'm talking about density, specifically); but if a planet has a low density, it will have to either be huge, or have low gravity. And at some point it will just be a gas giant.
(Indeed, the surface gravity of Saturn is 0.91g on the equator, and 1.23g on the poles. But you probably wouldn't want your world to be particularly similar to Saturn.)
Then (thanks to the other answer for reminding me of that) there's the effect fast rotation would mean for atmospheric circulation... no idea what it would be, to be honest. (Probably not much for reasonable values of "fast".) The only pages I could find regarding it only talk about Earth-sized planets (not larger ones), and - perhaps more significantly - planets fairly close to their star, which receive at least as much heat as Earth does, or more (as if that was the only possible position).
But it's certainly the best possibility so far.
A large moon can indeed reduce gravity under it a little. This is called tidal force, and if the planet is not tidally locked with the moon (i.e. the moon does not always stay over the same spot), will be very unpleasant to the planetary surface. (Just imagine regular tides magnified by a factor of several thousand - it will actually have to be even more than several thousand to get differences as big as what you're talking about - and you'll probably understand why. See also tidal heating.)
For the forces we're talking about, we really need the "moon" to be bigger than the planet, in other word, for the planet itself to be a moon of a larger planet (such as a gas giant). In this case, tidal locking is fairly likely, which means that the gravitational differences stay while the problems with extra-huge tides moving across the planet do not (as long as the orbit is circular enough); but even so, the gravity differences are tiny (this article seems to say they would be on the order of 0.001g or less for Io, a fairly close moon of Jupiter).
And if we try to make the planet closer to its "moon", for the differences to be higher, eventually those same differences will be enough to pull the planet apart; this is called the Roche limit. (Jinx must have been very close to it when it formed; I'm not sure it could have formed the way it is described while outside the limit, actually.)
It is, of course, very possible to just have some big mass concentrations (mascons), which is to say, large pieces of denser rock (or metal) near the surface. (This is what the other answer is talking about.)
However, they will not affect the gravity too much, because, if they're too thick (a bit over 10 km under Earth gravity), the crust will not be able to hold their weight, and they will just slide down into the mantle.
(As is fairly easy to see, the effect of a 10 km mass concentration will be only a bit higher than the gravity of a 10 km asteroid - that is to say, tiny. An example of a 10 km mass concentration is Mauna Kea.)
In principle (see again Jinx), if a planet is especially rigid, it could be possible for a mass concetration to be larger. But you'd need something almost the size of the Moon to get such a large difference, and such an object is very unlikely not to slide down into the core from the sheer pressure it effects on itself (this is the "not fall under its own weight" part of your mountain range argument).
As we have seen, all of the above methods except for rapid rotation (spinning faster) do not appear especially likely (to produce what you need without serious problematic side-effects).
So yes, to have a realistic planet, with a fully covering atmosphere, that has significantly different gravity in different parts, you need to agree to having a much shorter rotation period, and thus day/night cycle. (Not that shorter - about 8-12 hours.) And lower density. And since the density is lower, make the planet a bit larger, to have decent gravity.
In other words, something not unlike Planetocopia's Lyr.
The gravity of Lyr is 1.4g on the poles and 1.23g on the equator - not as much difference as you want, and the average is a bit too high (but then Lyr is 7 times heavier than Earth), but this is about the most developed such planet that I could find anywhere. (Other examples of large low-density planets in science fiction include Diomedes and Majipoor. I wasn't able to find detailed gravity figures for either.) So you might want a planet that's a bit smaller than Lyr (though then you'd have to explain why it formed with so little density), and rotates a bit faster. If your planet has a mass of 3-4 times that of Earth (with the same density as Lyr, or even a bit less dense) and a rotation period of 8-10 hours (compared to 12 for Lyr), you have a situation that is almost precisely what you described in the OP. (That is, if the maths for Lyr are correct, obviously. I hadn't rechecked.)
(But the day-night cycle would be 2.5-3 times shorter than that of Earth. Sorry. Also it will have relatively few metals - which is good to know when you'd be describing the world further.)
On earth already there is local variation in gravity, of course of a much lower magnitude that what you are suggesting.
This could not really have as much of an effect as you are suggesting. For a sizable effective force (about 0.1g difference), the planet would have to be spinning very fast indeed, which would have adverse effects on weather patterns. Several hundred mile-an-hour gusts would be the normal kind of wind on such a planet near the equator. A large moon would be unlikely to affect something on a small scale particularly noticeably.
So what causes it on earth?
In real life there is a very small effect of centrifugal force, but another contribution can be found over large mineral deposits. A planet with landscapes with huge, almost solid deposits of a heavy gravity difference you mention. However, any element that dense is almost certainly radioactive to some degree, so bear that in mind. Also note of course that sea level would be distorted by such a deposit, such that 'hills' of water could be found over undersea deposits.
Also, altitude can play a small effect. Higher up frames of reference can measure a lower effective gravity, and such effects have to be taken into consideration when using sensitive seismometer instruments