Statistics is the study of samples sets of data. Statistical study of data samples is used to make inferences about the entire population that the sample represents. For example, suppose you have a large order of French fries. After eating five fries and finding them salty and crisp, you might come to the conclusion that the entire order of French fries will also be salty and crisp. You inferred from your sample set, the five French fries that you ate, the properties of the entire population. You deduced that the whole order of fries will be salty and crisp. But what if all the salt is sitting on top of that order of fries? What if the fries in the bottom of the box are soggy?
Confidence intervals and confidence limits are a way to measure the confidence we have that the mean of a sample set is a true reflection of the mean of the entire population. For example, if the mean of your sample set is 50, we might have 95% confidence that the true mean of the population falls somewhere between 49-51, based on our analysis of the sample set. Typically, this is reported as a 95% confidence interval of 50 +/- 1. Your confidence limits in this scenario are a lower 95% confidence limit of 49, and an upper 95% confidence limit of 51. Confidence intervals and confidence limits can be calculated on a data set by any data analysis software, such as Microsoft Excel or Minitab.
Suppose you could measure the salt content of a French fry and record the data. After recording the salt content of your sample set, you can perform statistical analysis on the data and determine parameters such as a mean and standard deviation. But these parameters only apply to the sample set, not the entire population, as demonstrated by the example where the salt is sitting on top of the order of fries. If you measured every single fry in the entire population, you might find a vastly different mean and standard deviation than what you calculated for your sample set.
This scenario highlights the importance of sampling in any statistical analysis. You can see how the more French fries you test, the better your sample set will reflect the true mean, and the tighter your confidence intervals will become. If you test them all, then your sample set is equal to your population, and likewise, your statistical parameters will be equal for sample set and population. However, where testing (or eating!) every French fry in an order might be a feasible prospect, statistics is usually applied where the entire population cannot be measured. Doctors report reactions to new drugs based on a sample set of willing study participants. The more people a new drug can be tested on, the more accurately the results will reflect the reaction anyone taking the drug will have. In any statistical analysis, the more samples you test, the better your confidence in the numbers, the smaller your confidence interval, and the tighter your confidence limits.