I'm not a statistician. However, from what I remember of my biostatistics course I took in college, here is an explanation of the concept of the confidence interval. Statistics, and good experimental design, is about testing hypotheses. One usually states a null hypothesis and an alternate hypothesis. The null hypothesis typically states that there is no significant difference between two or more treatment groups. If a significant difference is found, then one can reject the null hypothesis and accept the alternate hypothesis. The latter typically states that there is a significant difference in some parameter between two or more treatment groups.
The following diagram depicts a normal distribution.
The numbers on the x axis indicate ± standard deviations. One standard deviation will be at the inflection point on the curve on either side of the mean. 95% of the data will lie between ± 2 standard deviations while 5% (2.5% at either end of the distribution) will be outside of this region. In statistics, alpha < or = 0.05 represents the point at which a calculated statistic is considered to be significantly different from the mean. In effect, the statistic will lie in the ± 2.5% of the distribution. When a researcher quotes P=0.001 or something like that, they are reporting on the probability of rejecting the null hypothesis. In effect, they are trying to persuade you that their data are "very significant." Some statisticians consider this to be nonsense even though it is the norm in peer reviewed research. Data are either significantly different or they are not. There is no such thing as "very significant."
By the way, all of those graphs you see in published research that use ± 1 standard deviation error bars to show that something is different are really demonstrating an investigator bias (i.e., trying to make the data look better than it really is). The error bars should always indicate a 95% confidence interval since that is what is used to determine significance.
Anyways, I hope this helps to clarify what is meant by a 95% confidence interval.