A bribe is any inducement or reward provided by Person A for Person B to act improperly in order to gain an advantage (whether commercial, contractual, regulatory or personal). A bribe may take the form of money, gifts, hospitality or some other benefit or a promise to provide one of those things in the future.
It is still considered bribery if it happens in a private business relationship (not just if public officers or government officials are involved). It does not matter whether the bribe is provided directly or indirectly and the person receiving the bribe is the same one who will deliver the advantage.
UK legislation counts improper payments for people to do their job properly (known as 'facilitation payments') as bribes. These typically involve a bribe to avoid an application or notice being 'lost' or put to the bottom of the pile.
It is also a specific offence under the Bribery Act 2010 to bribe a foreign public official. For these purposes, 'public official' is very broadly defined and includes anyone who:
holds a legislative, administrative or judicial position of any kind (whether appointed or elected) in a country or territory outside the UK
exercises a public function for or on behalf of any country or territory outside the UK or any public agency or public enterprise of such a country or territory
is an official or agent of a public international organisation