Still a problem with communication between the PC and the Nano.
Sketch uses 2080 bytes (6%) of program storage space. Maximum is 32256 bytes.
Global variables use 40 bytes (1%) of dynamic memory, leaving 2008 bytes for local variables. Maximum is 2048 bytes.
That part indicates that the sketch compiled properly and is ready to be uploaded to the Nano.
avrdude: stk500_getsync() attempt 1 of 10: not in sync: resp=0x00
avrdude: stk500_getsync() attempt 2 of 10: not in sync: resp=0x00
This says that the uploader is trying to contact the Nano and is not getting a response (failure to communicate).
Were you ever able to upload a sketch to the Nano prior to this?
Is any thing connected to the Nano besides the USB cable?
Are you sure that the right serial port is selected? Unplug the Nano and see what ports show in the Tools, Ports menu. What if any ports show up? Close the menu, plug the Nano back in. What ports show, now?