If I wanted to share files with someone and I had a static IP what is the easiest way to give them access over the internet. They dont have a static IP and I dont have a router. Can I just let them access my system using the IP in my network card?
Because you said "easiest", I'm going to recommend you use Dropbox to create a shared folder and then add your friend to said folder.
Short of using commercial software (even though Dropbox is free for the first ~2GB), you'll need to look into something like setting up a web server or FTP server to share your files which will get considerably more complicated and have security implications.