What is the screening process for Syrian refugees coming into the US?

The United Nations High Commissioner for Refugees does the initial investigation of the potential refugees and handles most of the preliminary checks. This information is then forwarded to all governments that have agreed to take refugees and have not exceeded their internal quota (which would include the United States). The United States Department of State then leads a second investigation of the potential refugee with the consultation of the FBI, the Department of Homeland Security, and several other federal agencies. The entire process takes over a year in length and far fewer than the maximum permitted number of refugees are actually settled in the United States.

It is important to note that the US process is far more rigorous than the refugee admission processes in Europe, which do almost no investigation at all into the supposed refugee's claims once they receive the recommendation from UNHCR or the migrant shows up on their doorstep.

This Article from "Time" explains the US process in much more detail.