There are two types of confidence intervals. One related to the mean of an interval variable and the other related to the percentage of a categorical variable (a nominal variable or an ordinal variable). Here only the former type will be covered.
Confidence intervals for means
The goal is to capture the true value/effect within some intervals. When repeatedly taking equally-sized random samples of a population, the mean of the samples will tend to get closer and closer to the population mean. The standard deviation of these sample means is called standard error. From the mathematical properties of the normal distribution, we know that about 68% of the sample means will fall within 1 standard error of the population mean. Thus, given a random sample of the population, there is a 68% probability that the population mean will be found within 1 standard error of the of the sample mean.
And similarly, from the mathematical properties of the normal distribution, we know that about 95% of the sample means will fall within 2 standard error of the population mean. Thus, given a random sample of the population, there is a 95% probability that the population mean will be found within 2 standard error of the of the sample mean.
These probabilities of the ranges where the population mean will fall are called confidence intervals. They show the probability of a margin related to the mean of a sample. Confidence intervals can be visualised with error bars. Summary: confidence intervals give us the estimated range of values for a population parameter, a precision estimate (indicated by the width of the confidence interval) and a statistical significance (if the confidence of interval does not cover the null value, it is significant at the 0.05 level – the null value is the value of a factor in the sample that is strongly thought not to exist in the population, such as a ratio of 1).
The effect sizes tell us how strong/large the difference/relationship between the relevant variables is. Even if you find a highly statistically significant in the difference/relationship between some variables, if the difference/relationship is weak, it is not relevant/valid/meaningful. Normally, measures of effect size like the Pearson’s coefficient take values (ignoring the positive/negative signs) between 1 (strong effect) and 0 (no effect).