Confidence Level (Certainty)
The certainty or confidence level is the probability that a result is
significant. A high confidence level means that there is only a very small probability that
the result (e.g. correlation or independence) happened purely by
chance.
Typically, the confidence level required is 95% or more: this means that there is 95% chance that the result of the analysis did NOT happened just by accident. Vice versa, it means that there is a probability of less than 5% that it did happen purely by chance. It is up to you to set the confidence level you want for a statistical calculation (default is 95%). In significance tables which allow you to interpret a results, the confidence level is usually marked as a "probability value" or "pvalue" such as "0.01" or "0.05":
If the data is normally distributed (bellcurve), the confidence level is indicated at the extremities of the "tails" of the bellcurve (see diagram below). The probabilty of a value happening in this area of the bellcurve is therefore very unlikely. The higher the level of certainty you set, the smaller the yellow area on the bellcurve... On this diagram, we have set the level of certainty at 0.05, which means that there will be no more than 5% chance that an event occurs within the yellow tails of the bell curve (ie: 95% chance that it happens in the white central area of the bellcurve). 
Confidence Limits (Margin of Error)
In
a research project, you usually collect SAMPLE data: a
sample of 100 people, 100 houses, or 100 measurements. You should
always estimate if the values you got in your random sample are
actually close to the results you would get IF you could measure the
ENTIRE population
(all individuals, all houses, etc). The confidence limits are the ranges
of values (min/max) within you can estimate, with a high confidence level (95% by default), that the mean of the entire population truly lies. The distance between the lower and upper confidence limit is the "margin of error" of the sample.
The confidence limits (min/max) is given by this formula, which uses the "margin of error": x = mean of the sample
z = zscore representing the size of the confidence interval you have set, measured in units of standard deviations from the mean (see below) s = standard deviation of the sample n = number of entries in the sample The element circled in red is called the "margin of error" of the sample What zscore should you use?
Formulation: "there is a 95% probability that the mean of the entire
population might lie within +/ the margin of error of the sampled
mean".
Example 1: estimate the mean IB Geography grade for all students worldwide, based on a random sample of 100 students
Example 2: polls
