You are probably getting an error response "Please Install IDENT."
If this is the case, then you probably need to configure your Firewall or Router/NAT to through-put TCP 113 traffic to your local machine.
If you are using Windows XP, it comes with a built in firewall you will have to configure to allow inbound connections on mIRC's IDENT and DCC ports. Read this tutorial on how to configure it properly - http://www.mishscript.de/ircguide/appb1.htm
(This tutorial recommends configuring a DCC port range starting at 4000... I would recommend a port range somewhere between 12000 and 65000. A higher and randomly chosen port range would make it more difficult for hackers to scan and predict these ports. IDENTd remains port 113)
If you are sharing your internet connection via Router or other NAT device, then you will need to follow a similar procedure on that hardware, usually called "Port Forwarding" or "Virtual Server".
You should also try the "Connection Issues" forum, as they probably have better answers to this problem.