What are the security issues of a computer connected directly to modem?

"Most modern routers come with something called Network address translation (NAT). This means that the 1 IP address that you expose to the internet is translated into the multiple IP addresses required for the PCs within your network. With NAT random incoming packets are not forwarded on to your PC, so if someone tries scanning your IP address and ports, all they will get back is rejection messages from your router (or in some cases, just nothing at all)

It's not by any means foolproof or perfect security, and I would still recommend a firewall on your PC, but having a NAT router between you and your modem is certainly a step up in security.

As a simple demonstration you can try this:
1) Find a firewall that logs blocked incoming connection attempts.
2) Connect your PC directly to the modem.
3) Watch the firewall logs start clocking up blocked connections.
4) Disconnect and attach a NAT router in between.
5) Again, watch the firewall logs and you'll see no blocked connections because the NAT router is blocking them all before they hit the PC."