Microsoft Access is a Relational Database Management System (RDBMS), designed primarily for home or small business usage. Access is known as a desktop database system because it's functions are intended to be run from a single computer. This is in contrast to a server database application (such as SQL Server), where it is intended to be installed on a server, then accessed remotely from multiple client machines. Microsoft (or MS) Access is a software package that you install just like any other software package, and is bundled as part of the Microsoft Office suite.
Excel may be fine if you've only got a small amount of data, and if you don't have many attributes against each piece of data. It may be fine if you don't have much in the way of relational data across multiple worksheets. Once you start storing many attributes against each piece of data, and perhaps you find yourself repeating information across multiple worksheets, then it's time to start using Access (or another database system if you prefer). Another important reason for using Access over Excel is, if you need to generate a lot of queries and reports. Access is much better suited for doing this compared to Excel.
When you create (and save) a database in Microsoft Access, the database is saved with a .mdb extension. This is the file extension you will use the most, when developing Access databases. Once you've established your database, you also have the option of saving it as an MDE file, which gives you some benefits over the MDB file. An MDE file uses a .mde extension.