First of all, people do not become resistant to antibiotics. Bacteria become resistant to antibiotics.

Basically, exposure to antibiotics can cause bacteria to evolve, as the bacteria that are not resistant are killed, and those that are survive. (Some bacteria have a way to share genes through a DNA structure called a plasmid, and antibiotic resistance can be passed along that way as well).

So the skin and the gastrointestinal tract are covered / filled with bacteria. If your bacterial microbiome has resistant bacteria in it, and you develop an infection from one of these bacteria, then the standard antibiotics might not work.

What does the doctor do?

Often, when treating a bacterial infection, the physician will obtain a test called a "culture and sensitivity". They take infected material (such as urine, pus, or sputum) and try to grow the bacteria that is causing the infection. After the lab grows it, then it can be identified through a series of tests (gram stain, for example). Once the bacteria is grown and identified, then they check sensitivities to antibiotics.

Sensitivity checking involves growing the bacteria again under new circumstances. Basically, a series of small paper discs impregnated with various antibiotics are placed on the Agar plate where they are growing the bacteria. By measuring the distance at which the bacteria can grown close to each disc, they can tell if the bacterium is likely to be sensitive or resistant to a particular antibiotic. Then the physician selects an appropriate antibiotic.