This is actually quite a broad question - there are a lot of ways for a BlackBerry device to interact with a server, and the ideal way depends heavily on your specific application.
Generally:
MDS Studio: the Visual Studio 2008 method is no longer actively supported. It was a mostly visual forms designer, but not a great solution for a lot of reasons. Basically, don't use it.
Mobile Web: Depending on what you need to do, this might work. Everything server side. The thing to be aware of (and why this doesn't work for a lot of problems) is that prior to BlackBerry 6 the browser wasn't very capable, either in terms of HTML support or JavaScript support. You can, however, create shortcuts to mobile web sites which appear as icons on the BlackBerry homescreen. And you can use BlackBerry Web Signals: http://na.blackberry.com/eng/developers/browserdev/websignals.jsp to push changes to mobile web pages to devices
Widgets: Supported in BlackBerry OS 5 and higher. http://na.blackberry.com/eng/developers/browserdev/widgetsdk.jsp These are packaged much like 'native' BlackBerry apps, but written using HTML and JavaScript. Can be a good option for rapid development if you're ok with just targeting OS 5, and with the limitations of the BlackBerry Browser. Lots of JavaScript hooks to system functions, so they give you more functionality than pure mobile web, and you can distribute them through App World and the like.
Java Apps: Most development work, but you get potentially the best user experience, and the most functionality. Too broad a topic to cover adequately here, but there are a lot of questions related to these on SO and elsewhere. A good starting point is the BlackBerry developer zone: http://www.blackberry.com/developers RIM Push and PAP Push both fall into this category, as they interact with Java apps, though there are other methods that you can use to achieve similar results (client polling, or I've even seen SMS or email used to push info or wake up a client, though this isn't recommended by RIM).
For a deeper overview of the options I'd recommend a book by my friend John Wargo, called BlackBerry Development Fundamentals. He covers the various options pretty well.