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.
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 is usually about more serious cases in which 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.
Examples such as these are clearly illegal (against the law) and would almost never happen in many Western countries. However in some parts of the world bribery is normal and it is difficult to do anything without offering bribes.
Bribery in some form is quite common in business in many parts of the world. 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.