A Juicebox gallery is a web gallery which can be displayed on a web page of its own or embedded in an existing web page (alongside other content) by following the instructions here.

A Juicebox gallery can be created on your computer using JuiceboxBuilder (a desktop application).

Juicebox-Lite (the free version) comes with JuiceboxBuilder-Lite and Juicebox-Pro comes with JuiceboxBuilder-Pro.
Please see the links below for further details on JuiceboxBuilder:
JuiceboxBuilder Tour: https://www.juicebox.net/tour/juiceboxbuilder/
JuiceboxBuilder User Guide: https://www.juicebox.net/support/juiceboxbuilder/

Juicebox-Lite (and JuiceboxBuilder-Lite) can be downloaded from here so you can try things out with the Lite version before purchasing Pro.
You might find the Getting Started guide useful.

There are also plugins available for creating Juicebox galleries with third-party programs (such as Adobe Lightroom).
Please see here for details.

Once you have created a Juicebox gallery on your computer, you would either:
(1) Upload the complete gallery folder to your web server and view the gallery's 'index.html' page in your browser (to view the gallery on a page of its own).
... or:
(2) Embed the gallery in an existing web page following the instructions in the first link above.

Just for reference, the Lite Embedded gallery is a Juicebox-Lite gallery using all default settings.
If you created a gallery with JuiceboxBuilder-Lite, you would not need to change any configuration options to achieve this layout and style.

No. Both Juicebox-Lite and Juicebox-Pro can be used in PHP or HTML web pages.
If you are embedding a Juicebox gallery in a PHP page, just make sure that you break out of PHP when inserting the embedding code.

There is a comparison chart showing the differences between Juicebox-Lite and Juicebox-Pro on the Download page.