The short answer is yes. Anyone on the same network as you can usually view your traffic by carrying out a very basic Man in The Middle (MiTM) attack like ARP Cache Poisoning. I cover ARP Cache Poisoning in a lot of detail in one of my blogs about Session Hijacking, which is another form attack that you're susceptible to here.
Also, as pointed out by others, WEP is a terrible security protocol and I give a demonstration of how you can bypass WEP security in 3-4 minutes in another blog here.
In terms of what he can see, he can always see the sites you're visiting. If the site uses HTTPS then he can't see the data you're exchanging, but if it's HTTP then he could see everything.
You also become vulnerable to a lot of other attacks like SSL strpping, DNS spoofing, code injection and others. Your network is a trusted zone and should be restricted to only clients that you trust. Not only that but if he uses your Internet connection to conduct illegal activity you could find yourself in some hot water. I suggest you upgrade to a stronger security protocol like WPA2 and change the key to something only you know.
I've written another blog showing how WPA2 is near impossible to break if you use a proper PSK here. If your router has WPS I would also disable that for additional protection as it present a weak point in WPA2 security as detailed here.