It depends of your network setup:
Almost all router have an integrated firewall, and you probably have one, so almost all ingoing network attacks will be blocked before your computer see them... So you must look in your router config for parameters and logs...
To monitor network devices there's essentially two protocols: Syslog and SNMP
If your actual router/setup is not sufficient, with the dd-wrt firmware you could get almost all professional router features (so, advanced logging) on a home router (if it's compatible and have enough cpu power and memory). (OpenWRT and Tomato are popular alternative too, but have less compatible device and/or are less user-friendly)
- And here are some windows clients/logs analysis tools for syslog and snmp :
- And some, more general, logs analysis tools:
(Server side software: built-in Windows Server SNMP service - list of windows syslog server in this serverfault question - linux built-in syslogd - or alternative daemon: rsyslog / syslog-ng (this one have a windows port, in Cygwin packages))
Nevertheless, if your network is partialy/totaly open or your computer is connected directly, and to monitoring outgoing suspect activities:
Comodo Internet Security is a good free firewall and it allows to log any matched rule.
(Online Free Armor & PC Tools Firewall Plus have received good reviews too, but I don't use them and I don't know about their capabilities to log)
(It contains also a very good HIPS (host-based intrusion prevention system) but if you want a dedicated product: Threatfire and Winpatrol have good reputation.)
More network inspection
As it was suggested by qwertyKid, you could make a step further with packets inspection and Wireshark is certainly the most powerful tool to do it.
(It's also a good idea to use a hub/smart switch to relay packets to inspect what's incoming in your network, but be careful, relay packets only to an unsensitive and well protected computer...)
Alternatively you could use more simple tools like those from NirSoft
- CurrPorts: displays the list of all currently opened TCP/IP and UDP ports and the process that use it
- SmartSniff: capture TCP/IP packets that pass through your network adapter, and view the captured data as sequence of conversations between clients and servers.
- IPNetInfo allows you to easily find all available information about an IP address: The owner of the IP address, the country/state name, IP addresses range, contact information (address, phone, fax, and email), and more.
- WhoisThisDomain allows you to easily get information about a registered domain.
More general security
If your computer is compromise, the network monitoring and inspection could be totally useless, so the first step to do is to secure it:
And as humans are the weakest part of computer's security, be careful at what you do...