Delta is the name of the Greek letter used in mathematics to represent "the change in". The partial derivative symbol is an invented symbol which, according to this Wikipedia article, is variously pronounced "del", "dee", "partial dee", "partial", "curly dee", or "dabba". The professor in the video pronounces it yet another way, doh.
The lower-case form of delta can be written with that vertical leg either curving back to the left, or with a kind of sharp 's' curve to the right.
I picked up the habit of curving my lower-case d's to the left when I took a biblical Greek class, because it was easier for me to distinguish my own written Greek from a lower-case sigma (σ). I still write my English d's this way, for absolutely no valid reason.
You can see some examples of both delta shapes here.
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