"Application handler" is an attribute that can be used within application forms once you've defined one or more options. (From the Setup panel select Application forms then the Handlers link). It's used when processing application forms for VSys Web and VSys Live primarily to indicate who in your organization is assigned each application. Application handlers aren't necessarily people, though, and this lets you use them for any purpose at all.
You'll see "Application handler" as a field in the Approve/Reject applications tool for VSys Live and the Process downloaded applications tool for VSys Web. (The field will only show if 1+ application handlers are defined.) Right-click on any application and select Application handler to change the handler for a given application.
This isn't a field on the application, meaning it's not something that the volunteer can see or edit online.
Each application has its own "handler" property. A single volunteer with two or more applications can have different handlers for each application.