In order to use replication, your "parent" database would be ArcSDE and your child replicas would likely be using SQL Server Express. In this setup, you would create an empty personal gdb on the SQL server via ArcCatalog. (In Database Servers, establish the connection to your SQL Server instance, then right click, and Create New Geodatabase.)
Then, you would use the "Create Replica" tool in the ArcGIS toolbox. Using this tool, you first select the feature classes from ArcSDE parent database that you want to replicate to your child database(s), and then select the child database to replicate to. This populates the child replica with the feature classes you selected.
Synchronization occurs only after you have made edits to your child replicas. Once you have performed edits to the replica database, you would use "Synchronize changes" (ArcCatalog, rt click the replica -> Distributed Geodatabase -> Synchronize Changes). This pushes only the edits to the parent ArcSDE database. From your description it read as though you understood synchronization backwards.