I gave my network security password to my neighbor so she could get online. How can I tell what she has been doing online or on my computer?
The solution is not for the faint of heart. What you need is something called Wireshark, a network traffic monitor, or some other network traffic monitoring tool. Such tools will give you the ability to look in and see exact request / responses, what sites the person visited, and even open up the data packets to see actual data coming down.
This will NOT work if the user is going to sites that use HTTPS (most email services such as Gmail, Yahoo, etc., or ecommerce such as Ebay), since all data will be encrypted.
Honestly, your best bet is to talk to the person directly if you are concerned about something. Make friends, be open and clear.
If you're worried about viruses coming through the network or something, you could just set up a separate subnet in the router / firewall for that person (or yourself), preventing network traffic from jumping across.