(This post is going to introduce a lot of jargon, and I’ll probably refer to it from future posts.)
Making a network connection to a computer involves an IP address (the computer’s address on the internet) and a port number. This is a flimsy analogy, but you could think of it like finding your way into a house: the IP address (the host) is like the street address, and the port number is like which entrance to use (the front door, or the window on east side).
For example, most web sites are on port 80 or 443: 80 is the standard port for web servers, and 443 is the standard port for a secure web site (HTTPS). When you want your browser to display the CNN homepage, your browser figures out the IP address for http://www.cnn.com (which at the moment appears to be 18.104.22.168), connects to port 80 on that host, and begins an HTTP transaction to download the home page.
A firewall is network software that allows or rejects network traffic according to a set of rules. An organization which hosts its own web site might have a firewall which allows the Internet to connect to the web server on allowed ports like 80 and 443, but the firewall would reject other inbound traffic to that host.
A port scanner is a program used to interrogate a host’s ports, looking for services listening on those ports. If you point a port scanner at a web server, it’s likely to tell you that a web service is listening on port 80 (and maybe also on port 443). Port scanners are powerful programs which can be used in many ways. A company might hire a security analyst who would use a port scanner to identify weak points on the company’s network (a good analyst would also give the company some suggestions about how to address those shortcomings). A malicious person might use a port scanner to the same effect but for a different reason: the port scanner can tell the attacker the ports on which services are listening on the company’s hosts, identifying targets he or she might try to compromise.
Sophisticated port scanners can identify specific software products listening on a port, and even the version of that software. If a cybercriminal used a port scanner to determine that a host was running version 2.2.15 of the Apache web server, he could use a search engine to look for vulnerabilities in that version of that product. He might find that that version has a race condition error resulting in a remote execution vulnerability and that someone has published an exploit which takes advantage of that software defect. He could download the exploit and run it against the web server for any number of malicious purposes (like stealing otherwise inaccessible information, sending spam and phishing attacks, etc.).
nmap has been a popular port scanner for a long time (it was first released in 1997). In The Matrix Reloaded, Trinity uses nmap to port scan a host she wants to compromise. nmap identifies a vulnerable version of a network service called ssh, and she then uses an exploit to elevate her privileges on that host. This is a good (and rare) example of credible hacking in popular media: the writer(s) included enough realistic detail to make the scene plausible.