the server part of a data base is where data is stored , manipulated and sorted/filtered for use in your end user program. having downloaded coldfusion I'm guessing that you are going down the route of a web based/type application. you don't need a database to make a web app/page work, but for some applications / web sites you may need to.
you would need a database storage engine if say you are allowing users to query data, narrow down their requirements. this site ferinstance uses a database to manage the threads and posts. if your site was whats called static (eg inforamtion that doesn't really change, doesn't have much user interaction adn doesn't provide a lot of data, then you probably don't need a db. Early so called "dynamic" sites used software like coldfusion to integrate the data with the website (the presentation bit).
the drivers mentioned are the mechanism of how an application talks to the database backend. all database products require a driver to 'talk' to the outside world. some times they are already ont he computer, most times they are installed along with the database or the application (in this case coldfusion).
if you need to use a data back end, then you don't neccesarily need to install a database on your own computer.. the database can be on any computer that your computer (and webserver use). its very common to have two systems (one for development.. this is usually on your own computer/s), and one for live.. ie where others can use the application. you do not want to be developing on a live website, although there is no reason why you cannot use a remote database. however as databases are potetnailly a security risk its relatively rare that an ISP will allow access to their database serveres from outside their security system.
talking of ISP's you need to decide waht your site is about, what it needs and based on that then go and find out what either your ISP can offer, what you can afford and so on. being an Apple user you may find the preferred choices are slightly different to other users. So lets assume you do need a database backend, you do need some form of scripting front end (coldfusion) then you need to find an ISP that supports both.
I suspect that MySQL will probably be the weapon of choice for your data storage (assuming you need a database). for whats its worth I think you will be better off downloading something like
XAMP which will install a db, web server (for testing purposes), various tools to interact with the webserver and database.
I'd suggest you also download the
GUI tool from the MySQL database
unless you have a confirmed reason to use coldfusion I['d suggest you consider a webscripting language like PHP, which comes as part of the XAMPP package.
there are good resources on the web and how to develop web stuff at
www.w3schools.com and of ocurse if you are suing PHP there is
www.php.net
good luck with your project