Nmap

nmap will port scan a computer and tell you what's open, it will also tell you other stuff, but that's the basics.

there's an online version here http://nmap-online.com/ if you don't have command line access to a computer that has nmap installed

nmap -p 1-5000 -T4 -sS 1.2.3.4	that'll get you started with a basic scan for IP 1.2.3.4
-sS	Syn scan, basically sends a signal to that box and sees if it gets any response
-p 1-5000	scans a specific port range
nmap -p 1-5000 -T4 -sV 1.2.3.4 --version-all	finds out what version of stuff is running on that server