Tried restarting the computer?
You might want to try restarting any router you have as well.
"Software caused connection abort" is something that the operating system's network stack is causing (though it can be triggered by external devices, for example I would get them when my router crashed. Rebooting the router fixed the problem).
If you have firewall software this could be a possible cause (including the windows firewall). Allow outgoing connections on TCP ports 6667 through 6669.
Yes, if you have p2p apps going this will easily get you to the maximum number of network sockets (there are such limits, in older versions of windows, I think it's as low as 64 sockets. Technical limits, has to do with the amount of memory the operating system reserves for network connection descriptors, although needlessly artificially low).
Lastly it could be possible that your ISP is blocking IRC, doubt this though because the message would generally be something like "Connection timed out" instead of "Software caused connection abort."
Horus and others: We do have CGI:IRC, which is a server based IRC client (you just need a webbrowser to access it -- your computer doesn't make a direct connection to the irc server), however I don't want to give out the URL / password to that unless you really can't connect (it consumes lots of system resources so I want to keep the users using it to a minimum).
