Pixel density is simply a measure of how many pixels will fit into a given area. It is determined by the size of the pixels: the smaller the pixels, the higher the pixel density.

Imagine making two mosaics of tiles on your wall: one uses small tiles, the other uses large tiles. They might end up looking something like this:

We can easily see the following:

1. Both mosaics are the same size.

2. The mosaic on the right uses smaller tiles: 4 times smaller than those on the left.

3. Therefore, the mosaic on the right contains more tiles: 4 times more than the one on the left.

Now imagine what we're really looking at is a highly-magnified view of two digital cameras' sensors. The "tiles" are the sensors' pixels (or light cells); it's clear that the sensor on the right has a higher pixel density than the one on the left. If we scale things up and imagine that the grid on the left is one tiny part of a 1 megapixel sensor (i.e. 1 million pixels), then assuming its sensor is the same physical size, the camera on the right will have a 4 megapixel sensor.

Now, we all know that more megapixels = better, right? :)

In fact, you hit on it in your question: megapixels measure image size but aren't always a good way to measure image quality. This is because smaller light cells tend to pick up more noise, especially in low light conditions. Camera manufacturers are continually finding better ways of combating this, but at the same time they're cramming more and more pixels onto the same-sized sensors. Very roughly speaking, higher pixel density tends to lead to more noise.

So to answer the last part of your question, higher pixel density doesn't mean better or worse per se. Increasing the pixel density of a sensor increases its resolution (more megapixels), which can be a good thing, but also usually increases its susceptibility to noise, which can be a bad thing. Buying any digital camera means weighing up those two factors and deciding on a compromise between the two that suits you best.