How does bacteria become resistant to antibiotics?

Bacteria becomes resistant to antibiotics mostly by the misuse of antibiotics.
For example, if a person is prescribed to take a certain antibiotic, say penicillin, for 2 weeks, many people would cease taking the penicillin after their symptoms disappeared, and would instead stockpile the antibiotic, hoping to save it for later, when it may need to be used again.
Contrary to common belief, does an antibiotic not destroy all bacteria at once, and it does not destroy bacteria randomly. The bacteria eliminated first are those which are most susceptible to the antibiotic, leaving those which are more resistant, and which carry resistant genes.
If a person only takes the prescribed antibiotic for a single week, as opposed to the prescribed two weeks, the more resistant bacteria are better able to survive and thrive, transmitting their genetic material, and thus their resistant genes, to next generations of bacteria, due to the lack of competition for resources after the elimination of the susceptible bacteria.
The resistant bacteria would eventually overrun the susceptible bacteria (if the trend continues), making the antibiotic obsolete, resulting in super resistant bacteria, such as VRSA (Vancomycin Resistant Staphylococcus Aureus).
Additionally, genetic material which provides for bacterial resistance may also be transmitted between different species of bacteria through a process known as conjugation, where bacteria form plasmids, basically capsules of genetic material, which are transferred from one bacterium to another.
There a couple of different ways this can happen. One is through our food supply. Due to the conditions that animals kept in, they are routinely given antibiotics to ward off disease. These antibiotics get passed on to us when we eat the animals but at a lower level. Pathogens in our body that would normally be susceptible to these antibiotics become immune to them because they can tolerate the low levels in our bodies.

Lastly, because bacteria can multiply at such fast rate, they can mutate at a fast rate as well. Occasionally these mutations make them more resistant to antibiotics and so we need stronger and stronger drugs to be able to fight them.