How Can You Reproduce and Understand the Meissner Effect Experiment?

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It would be helpful for you to provide some basic description of the content. Links to Youtube are too often the source of "joke" videos.
 
Well, one was the classic superconducting magnet experiment, the other a plasma light (think: neon sign), so start with the Wik links for those two concepts.
 
The first experiment is -as russ watters has already pointed out- just a demonstration of the Meissner effect using a type II superconductor (albeit without a spacer, they must have repeated the experiment a few times before they managed that, I am rarely that lucky), the liquid they are pouring in the white cup in the video is liquid nitrogen.
You can quite buy "Meissner kits" that consists of a piece of YBCO superconductor (the black block in the video) and a strong permanent magnet. Then all you need is some liquid nitrogen which is usually fairly easy to get hold of. You can buy these kits from e.g. Can Superconductors.