Methods for purifying plasmid DNA all basically depend on the concept that plasmids are small and that they are closed supercoiled circles.
Lysing the cells. The first step in all cases is to lyse the cells. There are various ways of doing this. The most gentle (and therefore the least likely to result in mechanical disruption of larger plasmids) is to incubate the cells with an enzyme called lysozyme. This breaks down the peptidoglycan cell wall of the bacterium, exposing the cell membrane. In the most gentle procedures, this is done isoosmotically in a glucose solution to avoid violent osmotic rupture of the membranes. A detergent is usually added to then added to disrupt the membrane. Alternatively, either boiling in detergent (usually Triton X-100) or exposure to strong base and detergent (usually a mixture of NaOH and SDS). The later two techniques are faster, but harsher and more likely to lead to mechanical damage of large plasmids.
Precipitation of the genome DNA and some of the protein. In the boiling preps, this happens naturally at high temperature. The proteins and linear DNA fragments from the genome, irreversibly denature and precipitate. For the alkaline lysis technique, the alkaline conditions also denature the DNA and some protein, which then precipitates upon addition of a large amount of salt and a drop in pH (both accomplished by addition of potassium acetate). In both cases, the plasmid does not denature and precipitate because the two strands are unable to separate in the closed, circular, supercoiled DNA. The ppt. is then removed by centrifugation.
Further purification of plasmid DNA. How you proceed from here depends on your goals. In many cases, it is sufficient to just ppt. the nucleic acid using either isopropanol or ethanol (isopropanol is more selective and thus normally used) and high ionic strength. If you need purer material you can perform an extraction into phenol or phenol/CHCl3 (the phenol tends to ppt. the protein and extract other contaminants), followed by a precipitation in ethanol and high ionic strength. If even purer DNA is required, there are two general routes available. You can use column chromatography (usually ion exchange). Many different disposable kits are sold for this purpose. Alternatively, you can perform a CsCl density gradient centrifugation. The later is a tedious affair in which you saturate the DNA with ethidium bromide, an intercalator. The amount of ethidium bromide that can intercalate into a closed supercoiled circle is different from that that can intercalate into a nicked circle and that is less than the amount that can intercalate into genomic DNA. Thus, the different forms all have different densities and band in different positions.
In bacteria, the isolation of genomic DNA is very similar to that of plasmids, except that after lysing the cells, you do not denature the linear DNA. Instead, you go straight to a series of phenol extractions and precipitations in ethanol. In eukaryotic cells, typically the outer membranes of the cells are subjected to digestion with a protease (protease K usually) that chews up the outer layer of protein. They are then often extracted directly with phenol and the nucleic acids precipitated with ethanol.