Where to find and how to use Ubuntu's version of Window's run utility

Where to find and how to use Ubuntu's version of Window's run utility

I'm switching over from Windows to Ubuntu, and I want to know how to get
the functionality of Window's "Run" command in Ubuntu. Usually in Windows
I just go to the Start button, and enter in the search field "run," and
the run program pops up. I mainly use this program to connect to servers
in my network by typing // + the name of the server. Here is an image of
the run command I'm speaking of

How do I replicate this functionality in Ubuntu? Do I use the command
line? I've read elsewhere about an old gnome program called "Run" that I
can get by installing the Compizconfig Settings Manager, but I don't know
if this is what I want.
For extra feel-good points, can someone explain to me what Windows' Run
command is doing under the hood? Is it just a shallow abstraction from
MS-DOS that run network connection scripts and other such types of
scripts?