After choosing which network (Mainnet, Ropsten, Rinkeby, etc..) to connect to on Augur- how does it sync the node with the Ethereum blockchain?
When Augur APP syncs what is actually happening is the "Augur Node" process contained within the app is ingesting a processing blockchain data.
The Augur Node process requests all Augur related logs events starting at the block the Augur contracts were uploaded at. When logs are received they are processed accordingly and alter a database that is stored on your machine. For example when a "MarketCreated" log is received the database will have a new row added for that market in the "Markets" table it maintains.
The Ui which is served by the app then makes requests to the Augur Node which uses the database it has built to serve those requests.