Protocols have standard ports. If you can access an E-Mail account right now (with one software application or website or whatever), and you want to access that E-Mail account another way (from another software application, or website or whatever), check the configuration details of the method that successfully checks the E-Mail account.
The "port" being referred to is a TCP port. The web page you reference provides common port numbers. Notice how many of the providers that use SSL will use the same port: port 995 for an SSL-based (POP3 or IMAP4) connection. Unencrypted communications often use port 110 for POP3 or, better (but less common), port 143 for an IMAP4 connection. (Interestingly, this port 143 wasn't on that web page that is hyperlinked in the question.)
If you're referring to web mail, then the typical ports are port 443 for an HTTPS site, and port 80 for an HTTP site. (However, web mail often needs a URL, not just a domain name. In fact, web mail is typically designed to be used by having a person manually go to the web page from a web browser. For what you're describing that you're trying to do, you probably will have more success by using a different protocol, like SMTP and either IMAP4 or POP3, preferably all over SSL.)
Ultimately, the information I'm giving you is just generalizations. The actual port that you need to specify is determined by whoever sets up the server you're contacting. (To be more technically precise, this usually refers to the firewall that protects the server you're attacking.) Since the port could be customized, the server could theoretically use a number other than the generalized standards. So whoever provides you with the E-Mail service should be able to tell you want port to use. Shame on them if they aren't able to do a good job of that.
However, there's really not much harm in just trying the common ports. As long as you're using encryption, and you're using a correct IP address (or DNS name), your private information (like your password) should typically go to the right computer. (I can imagine some weird theoretical setup where traffic gets redirected to another computer if you use another port, but that is not a common problem.) If you specify an incorrect port, your most likely result is simply that the connection to the E-Mail server will fail, so you won't be able to receive or send E-Mail. Fine. No big deal. So just try again with a different port number. There's not much harm in trying. The biggest cost will probably be that it will be a slight waste of time. That certainly isn't worth having you spend a lot of time trying to get an official port number just to avoid that cost. So, just try something. If it doesn't work, try another one of the common ports. You just may stumble across something that works right, and that will probably serve you well.
That linked list is pretty comprehensive. Any missing ones are impossible to give you since you haven't mentioned which webmail services you are using/need the port for. If your email providers can't tell you, then that's ridiculous, as it's required information to use their email service (unless its truly webmail-only, but again, we can't guess which services you are using). Please add more information. – Ƭᴇcʜιᴇ007 – 2015-02-05T17:54:14.270
In response to added comment about Britmail, [http://britmailnet.mail.everyone.net/email/scripts/serviceMenu.pl?user=new] indicates that access using common mail protocols (SMTP/POP3/IMAP4, SSL) costs $36.95 US per year. (Funny... BritMail.log has a British flag logo, but the prices quoted are in US dollars.) Looks like their strategy may have been to give people free web access, and then charge them when they want to access their E-Mail with more flexible methods. – TOOGAM – 2015-02-05T20:45:04.237
@TOOGAM thank you for your enlightening comment. – Ewan Delanoy – 2015-02-05T21:13:09.823