First we go back to Latin. The word 'prince' was derived from the Roman princeps -- one of the titles of the Emperors from Caesar Augustus on. (Augustus figured, rightly, that if he called himself 'king' or 'dictator' it might anger the Senate.) These Roman titles have quite a line of descent: Caesar has given rise to Kaiser and Czar (or Tsar). And in Medieval days, the Prince of Wales was the man in charge of Wales. (There are still people who mutter that if he'd called himself 'King' instead of 'Princeps' in the old Latin manner, the political evolution of Great Britain would have been different.) There are other ruling princes, e.g. in Monaco.
Be that as it may: the prince is either the man in charge, or the next in line to succeed him. And there's more than one way to claim the throne by blood descent. You can be child of the king -- or you can be the guy who came in, chopped his head off, spilled his blood all over the throne, and had a big enough following to make it stick. Rule by conquest, in other words.
Queens and Princesses have a similar situation. There was no doubt who was Princeps when Elizabeth I was on the throne. All you need to become princess is good luck and a good army. Descent by lineage is simply the peaceful way to do it. (And even then, it could get violent. Consider the Wars of the Roses. Richard III and Henry VII both had a decent claim to sit on the throne of England, and it took strenuous discussions on the battlefield to decide who got to sit, and who to lie silent.
For that matter, there was this little dispute in 1066 among Harald Hardrada, Earl Tostig, Harold Godwinsson, and William the Conqueror. But that was back when conquest was one of the standard ways to get a throne and a country to rule.