What do doctors mean when they talk about gram-negative bacteria?

Answer 1

To add to the wiki filled answer above, the gram negatives tend to be the more serious infections because the lipopolysaccharide layer is actually a toxin.

Answer 2

One of the earliest ways to detect and differentiate bacteria was a process called a Gram Stain (named after Hans Christian Gram, a Danish scientist).

This enabled doctors to determine which bacteria were gram positive (had an external peptidoglycan wall) from those that were gram negative (having an additional outer membrane protecting the peptidoglycan wall).

The gram negative organisms are known to be more virulent and in general they cause nastier infections.

Hmmm - the wiki list is somewhat deplete. These are some more gram negative bacteria. This list tends towards the pathogens as they are what come to my mind.
Campylobacter
Escherichia coli
Klebsiella sp
Neisseria sp eg. Neisseria meningitides (the dreaded meningococcus!)
Pseudomonas sp eg. Pseudomonas aeruginosa
Salmonella
Shigella