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The resolution, by itself, doesn't mean much. For a meaningful comparison, you need to look at one camera versus another, not one resolution figure compared to another.
Assuming you're talking about an APS-C sensor, 12 MP is high enough that the sensor resolution is only rarely the limiting factor on the resolution of the picture you get. In a typical case, the lens (and your camera technique) will reduce the resolution below what the sensor can (theoretically) produce. That being the case, a higher resolution sensor will only rarely help much.
At the same time, I should also point out that if you do have at least one really good lens and (for example) use a tripod at least part of the time, there's a fairly reasonable chance that when you use that combination an 18 MP sensor really will produce pictures with slightly higher "real" resolution than a 12 MP sensor.
It's also true that higher resolution in a fixed size of sensor is likely to imply smaller sensor wells, which will typically reduce light collecting ability. In theory that could lead to higher noise. Of course, there's a lot more to noise level than just sensor well size though. The difference in sensor well size is not enough to say with any certainty that a camera with an 18 MP sensor will have higher noise than a camera with a 12 MP sensor (in fact, if the 18 MP sensor is newer, noise might well be lower). If all else was equal, the 12 MP sensor should have lower noise, but all else is almost never equal.
To get a meaningful comparison, you really need to compare one camera to another, not one sensor resolution to another. Sensor resolution is a factor to take into account, but it's only one of many, and it's not (IMO) a particularly important one either.
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Edit: Okay, after supper tonight I had a few minutes, so I wrote up a simulator. I'll add a few pictures here:
Lens resolution exceeding sensor resolution:
Sensor resolution slightly higher than lens resolution:
Sensor resolution much higher than lens resolution:
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