router should be set up with dhcp ON, so it can assign different IP addresses to the pcs. both pc's should be set to obtain their IP addresses automatically. if the pcs have been set with fixed IP addresses, they will conflict and only one will get on, typically the first one that boots up, which sounds like your problem. hopefully you have access to the pc adapter settings and the router settings. if you must use fixed ip addresses make sure they are different on each pc.
note: even if the router is set to supply dhcp, it also may have a setting to force a fixed IP to a specific MAC address, which is unique to each pc. again, if used, this must assign different ip addresses to each device. also check your lease tables.
check your pc settings by opening an admin. command prompt and typing
then hit return. compare settings to same on working pc. incorrect dns settings can cause an apparent good connection without it being able to actually browse.
if the non-connecting pc is not picking up an IP correctly, windows assigns an APIPA (automatic private ip address) in the range 169.x.y.z. the router won't let that get on the internet tho it's OK for your local area network. sometimes right clicking the notification tray icon & choosing 'troubleshoot...' might fix that.
checking the logs on the router, and on the pc (with event viewer) may guide you as well if the above doesn't.