When we talk about "antibiotic resistant bacteria" we mean a strain of bacteria that is less susceptible to a particular drug or multiple drugs. This can mean that higher levels of the drug would be needed to treat it (which may be too toxic to give to patients), or, that no amount of drug can kill or stop the growth of this strain.

The ability to resist an antibiotic is generally something that is genetically encoded in the bacteria's DNA. Bacteria can become resistant by a few different mechanisms. Sometimes they do this using by making pumps that push drug molecules out of the cell. Sometimes they change the structure of the target that the drug is binding to, so it can't bind anymore. Other times they find a way to bypass the entire cellular pathway that the drug targets, while still doing their essential cellular functions.

So where does resistance come from? One way is through mutations. Every time a bacteria replicates and copies it's genome and sometimes makes mistakes. These mistakes can change the function of components of the bacteria, sometimes in a positive way, such as making it more drug resistant. If this happens in one cell, this cell will have a survival advantage over other cells, and will eventually take over the population. This is an example of evolution by natural selection!

However one reason that bacteria often become antibiotic resistance so quickly is that often the genes encoding resistance are already present in other bacteria in the environment. Many antibiotics that we use in medicine are naturally occurring compounds, and so over the course of history, many environmental bacteria have already evolved resistance to them. Scientists have found resistance genes occurring in samples of bacteria that are thousands of years old! These resistance genes can then be transferred between many different types of bacterial species, including the ones that cause human disease! This is much faster than waiting for the right mutations to occur.