Any statistical analysis requires data. The “sample size” is the amount of data underlying your statistics. It’s the size of your data sample.
Issues with sample size are usually brought up when (a) the sample size is too small for someone to reach a definitive conclusion, and (b) someone tries to reach a definitive conclusion anyway. That’s… uh, bad.
Let’s say you flip a fair coin, once, and it lands on heads. What’s your estimate of the probability of that coin landing on heads again? Well, based on your data sample it landed on heads 100% of the time (once, in one flip), so is it 100% probability? Of course not. The sample is too small for you to have observed all the possible outcomes and their relative frequency. Statistical analysis depends on what the data says, AND on whether the sample you’re using is sufficient to give confidence in what the data says. Size matters.