An effect size is the hypothesized difference expected by researchers in an a priori fashion between independent groups (between-subjects analysis), across time or observations (within-subjects analysis), or the magnitude and direction of association between constructs (correlations and multivariate analyses).
Effect size planning is perhaps the HARDEST part of designing a research study. Oftentimes, researchers have NO IDEA of what type of effect size they are trying to detect.
First and foremost, when researchers cannot state the hypothesized differences in their outcomes, an evidence-based measure of effect yielded from a published study that is theoretically or conceptually similar to the phenomenon of interest should be used. Using an evidence-based measure of effect in an a priori power analysis shows more empirical rigor on the part of the researchers and increases the internal validity of the study with the use of published values.
Sample size is the absolute number of participants that are sampled from a given population for purposes of running inferential statistics. The nomenclature of the word, inferential, denotes the basic empirical reasoning that we are drawing a representative sample from a population and then conducting statistics in order to make inferences back to said population. An important part of preliminary study planning is to specify the inclusion and exclusion criteria for participation in your study and then getting an idea of how large a participant pool you have available to you from which to draw a sample for purposes of running inferential statistics.
Due to the underlying algebra associated with mathematical science, large sample sizes will drastically increase your chances of detecting a statistically significant finding, or in other terms, drastically increase your statistical power. Large sample sizes will also allow you to detect both large and small effect sizes, regardless of scale of measurement of the outcome, research design, and/or magnitude, variance, and direction of the effect. Small sample sizes will decrease your chances of detecting statistically significant differences (statistical power), especially with categorical and ordinal outcomes, between-subjects and multivariate designs, and small effect sizes.
Statistical power is the chance you have as a researcher to reject the null hypothesis, given that the treatment effect actually exists in the population. Basically, statistical power is the chance you have of finding a significant difference or main effect when running statistical analyses. Statistical power is what you are interested in when you ask, "How many people do I need to find significance?"
In the applied empirical sense, measuring for large effect sizes increases statistical power. Trying to detect small effect sizes will decrease your statistical power. Continuous outcomes increase statistical power because of increased precision and accuracy in measurement. Categorical and ordinal outcomes decrease statistical power because of decreased variance and objectivity of measurement. Within-subjects designs generate more statistical power due to participants serving as their own controls. Between-subjects and multivariate designs require more observations to detect differences and therefore decrease statistical power.