Updating Craft cause broken git and merge conflict
The error that you are seeing is because you have changes in your local repo that need to be saved (committed) first.

I'm also a huge Git user and wanted to touch on a few items in your question.


The craft/app folder should always be in your Git repository as its the foundation of making your site run Craft.
If you want to see an example Craft Git project, you can take a look at this repository: https://github.com/themccallister/craft-foundation 
The only directories that should be ignored are storage (make sure you keep the actually directory using this trick: https://stackoverflow.com/questions/115983/add-empty-directory-to-git-repository) and any directories in public that will contain user uploaded files (i.e public/images/uploads). This will ensure that the directory contents are not overwritten during deployment, ensuring that backups, sessions and user content will remain after the deploy.


As far as your remote build... The typical workflow for Git should look something like this:


Download and update Craft locally
Commit your changes to your local environment (should mostly be updates in the craft/app directory)
Push your changes to your remote repository (typically referred to as origin or upstream)
Deploy the update.


Note: If you are building using a tool like (http://dploy.io) you can login and push the changes to your environment. If you are SSHing into the server and running git pull origin that would work as well.