How can I find open ports on a remote computer ?

In Windows XP/2k/2k3 you can use the PortQry command line port scanner to see listening ports on a remote computer. (If your installation does not have it included, it is available from the MS Download Center - see the links to Microsoft below).

PortQry> reports the status of TCP/IP ports in one of the following ways:

• listening - some service is listening on this port.
  
• not listening - port is closed, however portqry did receive response (ACK packet with reset flag for TCP, ICMP "Destination Unreachable" for UDP) from the target.
  
• filtered - portqry did not receive any response from the port.
  

Syntax
portqry -n name_to_query [-p protocol] [-e || -r || -o endpoint(s)]

Common command line switches:
-n [name_to_query] IP address or name of system to query
-p [protocol] TCP or UDP or BOTH (default is TCP)
-e [endpoint] single port to query (valid range: 1-65535)
-r [end point range] range of ports to query (start:end)

Examples:
To check a single port, use the -e switch:
portqry -n speedguide.net -e 80

For a range of ports, use the -r switch:
portqry -n speedguide.net -r 80:85


Note: PortQry also displays extended information for known services, such as SMTP, POP3, IMAP4, FTP, and is capable of performing LDAP queries.

See also:
MS KB 310099 - Description of the Portqry.exe Command-Line Utility.
MS KB 832919 - Features and functionality of the PortQry version 2.0
MS Download Center - PortQry Port Scanner.

And then there are GUI scanners... Check out NMAP, it's free, and one of the best out there.