That depends a lot on which drug they take. But in general, when a person is really addicted the common change that occurs in their behaviour is that they give up things that they used to like doing in order to spend their time getting or taking the drug.
Drugs are chemicals. In the short term, the majority of drugs give a sense of pleasure and enhance mood by interfering with normal chemical messengers in the brain called neurotransmitters. That’s why a lot of people start to take them.
Different drugs act by different ways. For example Nicotine in cigarettes is quite good at blocking hunger and appetite, Cocaine makes a person feel more confident and focused, Alcohol can relax people who are anxious, and there are many more. This seemingly positive short term effects are why people take them, but unfortunately drugs continue to alter how the brain is working long after one stops taking them. In people who have taken a lot of drugs they can even get withdrawal symptoms when they stop. A common example is big coffee lovers; a common withdrawal effect is headaches (the caffeine in coffee can be considered addictive).
So by disrupting what are brains usually do drugs alter our behaviour both in the short term (which usually is not too bad) but also in the long term.