Obviously, there are a lot of mod rewrite discussions and answers all across the web. However, I am having a hard time grasping them. So I thought I would ask here.
Here's something to get you going (put this inside your /.htaccess file):
Options +FollowSymLinks
IndexIgnore */*
RewriteEngine on
# if a directory or a file exists, use it directly
RewriteCond %{REQUEST_FILENAME} -s [OR]
RewriteCond %{REQUEST_FILENAME} -l [OR]
RewriteCond %{REQUEST_FILENAME} -d
RewriteCond %{REQUEST_URI} !/api
# otherwise forward it to index.html
RewriteRule ^.*$ - [NC,L]
RewriteRule ^app/. /app/index.html [NC,L]
NOTE: For newer Apache versions see also the next answer, which uses the much easier FallbackResource