Type I error rejects a true null (false positive), with probability alpha. Type II fails to reject a false null (false negative), with probability beta. Power is 1 - beta.
The trade-off is direct: lowering alpha raises beta for fixed sample size. Only more data improves both.
The practical point in finance. Power is usually terrible. Return data is noisy enough that most tests cannot detect economically meaningful effects, so failing to reject the null often means "not enough data", not "no effect". Reporting that distinction honestly is a research maturity signal.