Cigarette addiction happens when a person's body has a reaction to continual dosage of nicotine. It is partly an emotional reaction, building over time, to the use of nicotine-containing products. Someone addicted to nicotine may have a compulsion to use products that contain it, and that compulsion can come from both physiological and emotional processes. The addiction to cigarettes is just as severe as most other addictions.
Cigarettes are addictive because they have an effect on chemicals in the brain. When a person uses nicotine, it causes their body to release chemicals such as dopamine. That chemical, in heavy doses, can create a euphoric feeling in much the same way that the use of other drugs such as cocaine do. The addict's brain becomes dependent on that rush of dopamine, and they make a mental association between the nicotine and dopamine, which in turn leads to addiction.
When a person has this kind of addiction, their brain begins making less dopamine, so when they do not have access to nicotine, they will usually feel poorly. They'll often feel depressed, anxious and angry, and sometimes even lash out at those nearby. When the addict uses nicotine, their dopamine level is elevated, but just enough to make them feel "normal". Essentially, nicotine addiction makes one feel poorly all the time, and the addict needs nicotine to feel normal. The addiction happens slowly enough that the addict is often unaware that it is coming.
The addiction to cigarettes also has a psychological component; people become as hooked on the act of ingesting the nicotine as they do on the dopamine it produces. For instance, a smoker will usually link smoking with normal behavior such as watching a movie or reading the newspaper. When they quit smoking, they may feel as if they're missing something, and have a harder time enjoying their daily life.
There are multiple approaches to treating a cigarette addiction. Some people simply go "cold turkey" and deal with the withdrawals on their own. This approach is the most difficult and a lot of people can't handle it. There are drugs that dampen nicotine cravings, and there are also patches that allow the addict to gradually wean themselves off of nicotine for good.