A relational database organizes information into tables. Each row is a record, each column is a field, and the tables are connected through shared identifiers called keys. That structure is what lets the database enforce consistency, handle many users at once, and answer questions that reach across all of your data.
The "relational" name comes from the relationships between the tables. An order table can point at a customer table by its key, so the customer's details live in exactly one place and never have to be retyped. That is what kills the duplication and drift that plague spreadsheets.
Common relational databases include PostgreSQL, MySQL, Microsoft SQL Server, and SQLite. You usually never see them. They sit behind the business software you use every day, from your accounting tool to a full ERP.
For the whole picture, read What Is a Relational Database?