[Kalibasa]

I know that a node is where the wavefunction is zero, so I guess if the wavefunction squared is also zero, it's just a region where the electron is unlikely to ever be found. But I still don't really get what a node is- why do they exist? Or should I just not think about it too much?

I certainly don't know what they mean by "nodal plane", though; this is how they refer to hybridization in my freshman chem book, but they don't use that term in my organic chem book and I don't really get what they mean.

Thanks!

[azmanam]

Yes, a node is where the solution to Ψ² (square of the wavefunction) is zero.  There is zero probability of finding an electron at a node.

Maybe start here, see if it helps:
http://www.upei.ca/~chem342/Resources/Reviews/Molecular_Orbital_Tutorial.pdf