Uncertainty and error

In science, error is the unavoidable difference between a measurement and the true value of whatever is being measured. There is always error in every measurement, no matter how careful the experimenter or how elaborate the equipment. This is why it is important to take multiple measurements. It is unfortunate that the common use of the word error means mistake but in the context of scientific data, the word error does not mean mistake! 
Consider a measurement of the number of heads in ten flips of a coin. The table below shows the results of thirty measurements. Intutitively, you know the most likely outcome is 5 heads out of 10 flips. However, the outcome of a single measurement could be any number between 0 and 10. The most likely number is 5 but the data shows that out of 30 coin flips, 4 heads came up five times and 6 heads came up seven times.
Every measurement has variability so all measurements are uncertain to some degree
All measurements have some variability (error) that is random, like flipping a coin. One way scientists account for random error is to take the average of several measurements. To calculate the average, you add up all the measurements and divide by the number of measurements you have. The average of the 30 coin-flip measurements is 5.06 - very close to 5. The average is a better estimate of the true value (5) because averaging smooths out random variability.
Another reason for taking multiple measurements is they provide an estimate of uncertainty (or error). The solid red line on the graph shows the behavior of data that contains random variation. The difference between the average and the spread of the measurements is called the standard deviation. In the example the standard deviation is 1.33.
Measurements are always uncertain to some degree because they have random variability
The standard deviation is the uncertainty in a single measurement. The standard error in the average is smaller because many measurements are included. The standard error is the standard deviation divided by the square root of the number of measurements. A proper result states the average plus or minus the standard error.

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