Antiobiotics kill bacteria but bacteria can become resistant to antibiotics. This happens through a process called natural selection. Within a group/population of bacteria they will all have small differences because of random mutations. These mutations can make them stronger or weaker, resistant or non-resistant to antibiotics. So when you give an antibiotic those with a mutation that gives them resistance survive and the others die. This selects a small population of resistant bacteria that can then grow with less competition. The whole population is now resistant to this antibiotic.
An example of this is MRSA (Methicillin resistant staphylococcocus aureus). It is resistant to a lot of antibiotics making it very difficult to treat and therefore dangerous.
To avoid bacteria becoming resistant it is important to avoid giving antibiotics unecessarily and to complete the full course.