As far as I am aware, you can't directly access many major database technologies directly from salesforce (ie. send a query string to mySQL), but what you can do is make web service callouts to a service that exposes your database information? Without any details on your database/technology setup, I can't advise any further on that, but if you can/have exposed your external database via a web service, I'd recommend using RESTful call outs to access it from Salesforce.

Basically you will want to create a new Apex class representing your call out functionality, and use the built in Apex from WSDL request library to construct a webservice call out (much like the one for the Amazon example) or you can roll your own.

I prefer to write my own, and here is some starting infomation on this process..
https://developer.salesforce.com/page/Apex_Web_Services_and_Callouts#HTTP_.28RESTful.29_Services

You can set all your header and authenticaltion informaiton using the built in HttpRequest library (http://www.salesforce.com/us/developer/docs/apexcode/Content/apex_classes_restful_http_httprequest.htm#apex_classes_restful_http_httprequest), and then make your call out with http.send(req);

Which returns your HTTPResponse object, that you can then parse. Read
https://www.salesforce.com/us/developer/docs/apexcode/Content/apex_classes_restful_http_httpresponse.htm
for more information on that.

If you don't yet have a webservice exposing your external database, you have tonnes of options in the computer science arena(!!) you could use a simple PHP script on a web server, or a complex Java application. You can also use loads of integration tools to expose your database information to salesforce, including free ones (like Talend) and paid ones (like Jitterbit).