Chlorine is a reactive chemical, quite like oxygen actually (actually not as bad as oxygen), but our bodies are designed to deal with oxygen so we don't get poisoned by it. What chlorine does is it corrodes moist surfaces (like the inside of your lungs) by getting into the water and forming acids. If your lungs are dissolving from the chlorine, you won't be able to breathe, and that's how it will kill you.

The quantity of water in a swimming pool is so large that the chlorine is diluted to the point where it is only very mildly toxic. Moreover, as a human being, you're breathing air even when swimming and the chlorine can't get through your skin so easily (it's why we have skin, to keep poisons out). Likewise, you're not drinking the stuff. Still, it's toxic enough that if you're swimming a lot with your eyes open; your vision will become blurry for a few hours afterwards. An animal that did breathe it (like a fish) would die, and of course the reason why we put chlorine in swimming pools is to kill the algae that would otherwise foul up the pool!

Youre right- chlorine gas (Cl₂) is not very good for you.However, in very small amounts, chlorine gas will not be fatal but will just irritate your eyes and lungs temporarily. The chlorine that is added to your swimming pool is in the form of ions, which are dissolved in the liquid rather than in a gas form that you can breathe. As ions, the chlorine can kill bacteria and microorganisms. However, even in dissolved form the chlorine added to pools can have some side effects on humans. You might have noticed after swimming in a pool that you have itchy or dry skin, or maybe red eyes if you tried to open them under water. This is why chlorine is added to pools only in very small amounts. If you added too much chlorine to a pool, it would start to be bad for people. But it is much safer to add a little bit of chlorine to the pools than to swim around with all the germs that would exist without chlorine.

Chlorine dissolves in pool water to form other compounds and generally does not remain in the chlorine gas state (in fact, it's most often not injected into pools in the gaseous state, either). The main reason why chlorine gas is potentially fatal is that it is heavier than air, so when it is released, high concentrations of it stay close to the ground where people are trying to breathe.