There are two definitions of NP I found:
1) NP is a set of problems that have poly-size certificates, and with a given input, there is a poly-time certifier that checks the proposed solution.
2) NP is a set of problems that can be solved by non-deterministic algorithm in polynomial time.
I think (1) is a standard definition that appears everywhere, but I found (2) from one of the lecture videos, but to the best of my knowledge, I think the reason why they're equivalent is that, a non-deterministic algorithm can make a lucky guess out of polynomially many options in constant time, and a result of these guesses would be our certificate.
But why (2) is restricted to guess one out of polynomially many options? Why can't it guess one out of exponentially many options? I don't have any knowledge about how non-deterministic algorithm or turing machine works.