If an infection comes back a few weeks after taking antibiotics, because the full dose and prescribed amount of antibiotics wasn't taken, does this mean that the bacteria is now immune to that antibiotic? Or can it still be treated the same way?
First of all, nothing is immune to anything, just resistant, and resisitance itself varies. If you develop a bacterial or viral disease, your body will develop antibodies against it. This makes you resistant to the bacteria or virus, but not immune. If you are exposed to a sufficient quantity of that same agent, you will develop the disease again. This is true for antibiotic-resistance, as well.
So...when you take antibiotics for some bacterial disease, some small percentage of those bacteria may be naturally resistant to the bacterium simply because of genetic mutation, may have developed strategies that allow it to mutate in response to environmental stimuli, or have borrowed it from cross-breeding1. When you start taking the antibiotic, it kills off the non-resistant bacterial population first, which is generally the vast majority if it. Normally, you are prescribed a sufficient amount of antibiotic to kill off the whole bacterial population. The problem arises when people stop the antibiotic regimen because they're feeling fine and see no reason for taking the remaining prescription. This leaves behind the resistant bacteria. And if your immune system doesn't take care of the rest, you will redevelop the disease, and it will then be resistant the that antibiotic. Not only that, but you will then spread that antibiotic-resistant bacteria to others.
Depending on the strength of the bacteria's resistance the disease may nor may not need to be treated with a different antibiotic, but most likely it will. If treated with the same antibiotic, the dosage and/or duration will need to be increased.
Here's a decent article.
Note: by mentioning viruses I do not mean to imply that antibiotics may be used to treat viral infections. Also, I omitted bacterial resistance developed by other means (e.g. farm animals, food sources) because it seemed beyond the OP's question.