The A260/A280 gives you a sense of the relative amounts of DNA, RNA and protein in the sample. DNA has a A260/A280 ratio of 1.8 while RNA has a ratio of 2.0. Thus, A260/A280 closer to 2.0 than 1.8 suggests significant contamination with RNA. A ratio significantly lower 1.8 than this indicates contamination with protein (which absorbs light at 280nm). Note that none of these checks give you good quantitative information about the degree of purity.
On an ethidium bromide-stained gel, you would only be able to visualize DNA and RNA as ethidium bromide stains only nucleic acids. Thus, you cannot assess contamination by proteins using a standard ethidium bromide-stained gel. However, the gel can give you an idea if there are any other DNAs or RNAs in your sample (something the spectroscopy approach above cannot tell you). Thus, these two methods provide very complementary information about the purity of your plasmid DNA.