It would help to know what websites you can't access.
You say you can ping the websites - try using WGet and see if you can retrieve the pages from those desktops.
Furthermore, you said some computers can't, which leads me to assume that others on the same network can (and your laptop would be evidence of this). In which case, you have one of two possible culprits:
1. Software configuration on the computers themselves - ESPECIALLY since you said ping and tracert both work, it points to the browser. Like I said, try WGet first. If you can fetch pages through it, then try Firefox. The likely culprit is Windows security settings, Windows firewall settings, other software firewall settings, or antivirus settings. Sounds like the browser is getting blocked, but some traffic isn't. WGet will give you a better picture of that.
2. Your router is filtering traffic to specific internal IPs. This would explain why your laptop can get out, but some desktops can't. This one is easy to test - unplug a desktop, assign its IP to your laptop, and try the pages again.
All in all, a bizarre problem.
Also...
We need a better description. How many desktops on site, how many can't access certain webpages, and are the webpages universal to all the problem computers, or does each have its own problematic few?