How to copy files from one Mac to another using a ‘router’ and ‘directly connecting them’?

I just bought a second Mac laptop and want to copy big files (pictures, music, videos) from one to the other. I have a 4 port router and have connected them both to the internet but one can not see the other.

And, I also want to know how to make one Mac laptop appear as a ‘drive’ to the other.

Is there a security risk using either method?