Bribery means offering something (e.g. money) to a person in return for some favour which is bad in some way.
The money that is offered is called a bribe and the verb is to bribe. "Active Bribery" is offering payment and asking for favour, and "Passive Bribery" is asking for payment and offering favour. It's still called bribery, even if the trade is never done.
It is sometimes difficult to decide whether something is a bribe or just a reward. If a father pays his son for washing the car, this is just a reward or payment. But if a parent pays a child for eating up its dinner this might be thought of as a bribe, because most people would think this was not right.
Bribery can be a crime, in more serious cases, like when a person offers money so that he does not get into trouble. If a motorist is caught speeding by a policeman and he offers the policeman money or a bottle of vodka to persuade him not to fine him, this is bribery. If someone wants to take something into a country that they are not allowed to take in (or that they would have to pay tax on) they might offer the customs officer a bribe to persuade him to let them through. Some corrupt people will not do their jobs at all (like delivering some package) unless they get a bribe on top of their normal pay.
Examples such as these are clearly illegal (against the law), and are rare in many countries, but bribery is quite common in business in many parts of the world. It is difficult to do anything without offering bribes in these places. Giving business people presents in the hope that they will want to do business with you may just seem like good manners at times, but in some cases it may seem more like bribery.
People who are found to be taking bribes can sometimes lose their jobs. In some cases (like the case of the motorist) bribery is against the law.