"shell" here generally refers to a type of access you have to your DH server. If you're running windows on your home computer and if you've brought my a MS-DOS prompt that's basically what this is. Shell access gives you a command line text interface with your server. This give you a great possibility of things to do.
From your computer there's two "protocols" you can use to connect to the shell of your DH server. The preferred method is SSH and the other option is telnet. SSH is secure (similar to SFTP) and telnet isn't (similar to FTP). I know you said you didn't want a wiki article, but the wiki artcle on SSH is fairly helpful. But here's the basics from that.
You can use a program such as Putty on windows to connect via SSH to your server. (that's a direct link to the download page - just select the win 32 version). You can also use the build in windows program Hyper Terminal (Start > All Program > Accessories > Communications) to connect via telnet.
But, here's a small snag. Before you can log into your server with one of those programs your user has to have 'rights' to do that. You'll have to log into the panel > Users > Manage Users. Edit the user in question and place a check mark for SSH access. This change normally takes place in a min or two, but may take upward of an hour or two.
Hopefully that makes things a little more clear for you. As for why it's not such a simple process here at DH as your previous host... I don't know. It's just not a feature that DH has. Generally they encourage people to use their own default setup, but the possibility is there no only to have your own php.ini file but to make your own install of php. That sounds like a hand suggestion though, to be able to just place a php.ini file in your folder and have it used - you might consider submitting that idea to DH via the Panel > Home > Suggestions. Once put in place others can vote for the idea to help give DH a feel for what features are most important to their users.
You can also install several other such things with your own configurations and what not - as long as it meets a few basic requirements. First it can't be a persistent process - it can't constantly run on the server regardless if someone's using your site or not. For instance things called "severs" can't be run on DH. Next you have to be able to configure the install to put all the files in your home directory as you don't have access to the root folders. And it also has to play nice on a shared server and not suck up to many other resources. You probably don't care about all this right now, but I was just trying to entice you with the greater possibilities.
Please post back if you have more questions. I know how frustrating things can be when you're getting started with a new host. We've all be there.
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