To understand the "Galactic Plane" explanation, you need to think in 3 dimensions. The galaxy is a flattened disc of stars and the solar sytem is rotating round it in a period of about 225 million years. As the solar system orbits round, it bobs up and down by a small amount. rather like a roundabout at a fairground where you can sit on a horse and bob up and down as you go around. The period of this up and down motion is about 60 -70 million years. Studies show that we passed the mid-point in the vertical direction in the last 3 million years and therefore won't reach this point again for another 30 million years or so. There is a diagram here, although the vertical excursions are greatly exaggerated: http://the2012deception.net/?p=15
Moving on to the "Solstice Alignment" page, this is purely a line-of-sight effect, with something lining up with something else as we see it in the sky. This is essentially in 2 dimensions, with objects in the sky appearing to pass in front of one another even though, in reality, their distances from us are vastly different. The galaxy, or Milky Way appears to us as a band of light all around the sky. As we orbit the Sun, the Sun appears to circle the sky and it crosses this band of light twice a year, in June and December. In fact, the centre of the galaxy is 2 billion times farther away than the Sun and there's no physical significance in the fact that the Sun appears close to it every December. Due to the fact that the Earth's axis is slowly changing its direction (see Precession: http://www.2012hoax.org/precession) the date on which this alignment occurs has been gradually changing over the centuries at a rate of one day every 72 years. It just so happens that at present, it occurs on or close to the dates of the solstices (about June 21st and December 21st). The date on which the solstice point was calculated to exactly coincide with the mid-plane of the galaxy was 1998. So the alignment in this case is that between the solstice point (defined by the Earth's axis) and the galaxy.
Sorry if this appears complicated, but the proponents of doom in 2012 tie themselves in all sorts of knots in order to extract a small snippet of something they think is rare or unusual, but which in fact is routine and uneventful.
Feel free to ask again if anything is still unclear.