Clipping/exporting multiple selected vector layers in map to extent of another raster/vector layer in ArcGIS Desktop?

I think Batch Processing is the simplest way to go.

1. Locate the Clip tool in the ArcToolbox window (Analysis Tools toolbox, Extract toolset) where you can right-click on it and choose Batch. (Note: You could do this from the Search window at 10.1 but not 10.0.)
2. Multiple select your layers and drag them in as Input Features
3. Select the polygon layer you want to clip with and drag that into the top cell of the Clip Features, then right flick and choose Fill to copy to the rest of that column
4. Hit the Environments button and set Current Workspace to the geodatabase or folder you want to output your feature classes into - this should populate the Output Feature Class with names based on each input feature class
5. Hit OK to run