In a nutshell, here are the definitions for all three.
1. Significance level: In a hypothesis test, the significance level, alpha, is the probability of making the wrong decision when the null hypothesis is true.
2. Confidence level: The probability that if a poll/test/survey were repeated over and over again, the results obtained would be the same. A confidence level = 1 – alpha.
3. Confidence interval: A range of results from a poll, experiment, or survey that would be expected to contain the population parameter of interest. For example, an average response. Confidence intervals are constructed using significance levels / confidence levels.
Read on for several examples and more elaboration.