There isn't an exact word in common use. We just say 'tip'. I'm as interested in fingernails as most people, but I've never heard of either of the terms which Ehsan suggests. 'Distal edge' is extremely technical, and I always thought 'free margin' was to do with economics. Nobody would know what you meant if you referred to your nails using those terms.

There is no commonly used term for this part of the fingernail. You can say "I am cutting my nails" without specifying which part of the nail is being cut, everyone will understand what you mean. There are specific technical terms for the various parts of the nail, but the only colloquial term you are likely to hear in reference to nail anatomy is "quick", which is where the nail joins onto the finger: "She chewed her nails to the quick".

free margin or distal edge

There's not really a common word for it that everyone uses, but you could say either 'finger nail tip' or 'edge of the fingernail'.