Sometimes it’s obvious (e. g. olinguito, a big furry mammal we’ve never seen before.)
With other species that are closely related (visually) to existing ones, it’s generally various bits of body measurements (morphology), including (and especially) skeletons if possible.
Another way to do it is DNA sequencing… you can generally tell how distant two groups of species are with a little bit of comparing their DNA. This is currently done in conjunction with the morphological studies I mentioned above.
However, there really are no hard and fast rules for where you would draw the line, and there our countless nitpicky arguments ongoing for various closely related groups, and what constitutes separate species (which can sometimes bleed into the comfortable / political realm.)
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