MSDN should have some articles about this feature, we also have a chapter about Proactive Caching and chapter about Processing in our book Microsoft SQL Server 2005 Analysis Services , which explains how processing and proactive caching work and gives some recommendations on how set them up. I would recommend to read about this feature and research it a bit to find the right strategy to update your data, based on the update patterns in your system.
This is quite complex feature (IMHO) and while it seems easy to set up in the user interface you should be careful, so you won't slow down your system too much and not cause too many updates. On the other hand you can use Analysis Services feature called Proactive Caching, which enables you to set up Analysis Services so it will try to reprocess dimensions or/and partitions when change in the relational database is detected. If you are just adding members to the dimension you can use ProcessAdd option of the Process command, or you can use other options (I believe they are covered in the whitepaper that Tomas recommended). As you noticed the whole cube doesn't have to be reprocessed. By default, Analysis Services is pretty isolated from the relational engine, it means that any changes to the relational table won't be detected and AS server won't be updated until you manually issue the command Process, which loads data into Analysis Services. I would also try to explain a little bit how processing in Analysis Services works. I would like to stress Tomas's warning about converting dimension into the ROLAP mode, which will dramatically decrease performance of your system.