According to the British Computer Society's glossary, a database "is a (large) collection of data items and links between them. structured in a way that allows it to be accessed by a number of different applications programs".
At a very simple level, a database can be a collection of names and their associated phone numbers, like you might have on your mobile phone, or it could be the details of members in a local club kept in a book. It could be a larger set of details, like you might find in the Yellow Pages. It could be a collection of records about patients that are kept in a doctor's surgery, or a set of tax records for all of the companies in the UK, or the records about criminals and crimes that the police keep, or details about the stock kept in a supermarket and so on. In fact, when you start looking, databases are absolutely everywhere and used by many different kinds of people regularly.
Databases originally were all paper-based. Each piece of information about each person, or a club member, or a company, or a product, for example, was known as an 'attribute'. All of the attributes together (all of the details about a person, a company etc) were known as a 'record' and were collected together and typically written on one card. The record cards were then collected together to form a 'file'. This file was then put into a filing cabinet. Users would then go into it to find information in the records. A 'database' was simply one or more files, so for example, you might have a file for animals' details, another for vets' details, another for all of the zoos in the country and so on. All the files together (although there might only be just one) were known as a database.