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I do know that if two animals cannot naturally reproduce, they are said to be of two different species. But what determines the genus?
Just for example, the African elephant belongs Loxodonta but the Indian counterpart belongs to Elephas , despite the fact that they are so similar, as opposed to say, the fact that, the Bengal Tiger and the Snow leopard despite being so different , are placed in one genus, Panthera ?
I don't think there is any non-arbitrary concept being used to tell whether two species belong to the same genus or not. The same goes on with any other taxonomic rank. Systematicists build some intuition within each clade about what a genus could mean and use it to argue what is the most strategic grouping but the concept of genus may well vary a lot from clade to clade (from spiders, to mammals or to flowering plants, ...).
Note that even the concept of species is not as nicely defined as you may think. Have a look at How could humans have interbred with Neanderthals if we're a different species? for discussion on the concept of species.
What also SciShow forgot to mention is that, by the very definition of evolution, taxonomy is just a handy way that we invented to simplify our life (or complicate, if you ask me). As they pointed out, categorizing and identifying appeals to our brain. But it's just a photo of a runner; if you snap it a few seconds (or geological eras) apart then you get the runner in a different position.
Reminds me of a riddle I was told some time ago: "where was the men when he jumped off the bridge?". If your answer is "in the air", well that's AFTER he jumped... likewise "on the bridge" is BEFORE he jumped.
So what we today identify as a specific species or genus might very well change in the future decades or even years and taxonomy is full of cases in which species where "relocated" and renamed. Sometimes changing the parameter on which you build the taxonomy (mostly now is gene based, before it was morphology based) might completely turn the tree upside down from the roots. Plant taxonomy (being a botanist that's what I'm familiar with but I suppose it happened also with animals) was revised multiple times because some clades would change position depending which gene was used as reference.
