cPanel comes with RoundCube, Horde and SquirrelMail.
Of those 3, my favourite is RoundCube (which in this context is a webmail interface rather than an app). I find it to be a pretty decent webmail client. Try and see what you think. It's improved a lot over the past year or two.
https://yourdomain.com:2096 should get you to a sign-in, and from there you can choose which webmail interface you use. Roundcube should be one of the ones listed.
No - if you use the port 2096 link I gave you, you'll be able to enter the e-mail address for the mailbox and the password for that mail account. You can gain access to that one mailbox without accessing cPanel.
I have just set up an email address in cpanel and now want a web based email client. It needs to be web based because there are two different people who need to be able to access this email client.
Any recommendations? Free ones!
RoundCube works well for a webmail based solution. But also remember that you can set multiple users up to access the same mailbox using IMAP. They will have access to all of the same folders and messages using IMAP. Try the Thunderbird email client in IMAP mode (instead of POP3). A good IMAP based email client will nearly always work better than any webmail client.
As mentioned each cPanel comes with Horde, Squirrelmail and RoundCube by default and your client should be able to choose which of the three they want to use. Its only a case of personal preference as all 3 have practically the same functionality, just different outlook.
To be fair tho, there are minor differences that may make one or the other more favorable - SquirrelMail is the most basic of the 3 with its simple old school design. RoundCube is a little more easy-on-the-eye but in terms of functionality its practically the same. The only one that differs more is Horde - it benefits from a mobile device interface, more options for task organizing, calendars etc.
At the end of the day, each of them can easily serve its purpose and they are quite reliable too