Suppose we want to find the amount of variation of marks scored by 60 students of a class in which I teach, then we have to find standard deviation. Here total number of students are very less and we have actual data.

Now suppose we want to estimate the average life of battery. Clearly, from several point of views, it is not possible to collect the entire data. So, we have to take samples. Suppose we take 100 samples with 10 batteries in each sample. We find the average life of each sample. That is, we find the mean of 10 items in each sample. There will be 100 such means. Clearly these means will not be exactly same as the mean of the entire population. Standard error is related with the standard deviation of these 100 means. You can refer the following link for a clear idea.

https://en.wikipedia.org/wiki/Standard_error

Similarly, when the results of a series of measurements are described by a normal distribution with standard deviation sigma (Greek letter) and mean 0, then erf(a/(sigma*sqrt(2))) is the probability that the error of a single measurement lies between −a and +a, for positive a. Thus error function represents a particular probability. Standard deviation and standard error can be any positive number. But the value of the error function will always be less than 1 as it is a probability. Again, for a better understanding, refer the following page:

https://en.wikipedia.org/wiki/Error_function

Thanks for the answer. I see now the difference. However, a new question arises, why is not erf more frequently used to report variation in neuroimaging?

Since erf is representing the probability that the parameter of interest is within the range between -x/ std * sqr2) and x/std * sqr2) ( i.e: it is inside the  "accepted deviation" -not outlier) i consider it more informatif to determine extreme scores. Given that in neurogimaing it is relatively likely to record extreme scores, i do not see why we do use it more often..

Thanks again for your time

Note that the sigma used in the definition of error function is the standard deviation of the normal distribution which fits with the given measurements. The reason that erf is not used more frequently may be that the standard deviation of the population may not be always known. Another reason is that we also must have tables to find its value. Of course, I am not sure about it.