I feel like people might be overcomplicating this. If you take a sample from a population, you get two main statistics from it: The mean, and the deviation. One describes the center of the data, the other the distribution around it. Imagine you kept drawing new samples again and again. You can make a list of the means, right? They should all be fairly close, but the random sampling means they're all slightly different. That list of means has it's own mean - and it's own deviation. That deviation is the standard error of the mean. It's a measure of the distribution of means in many samples of the same population. Now, the formula you're probably familiar with obviously doesn't draw many samples from the population! It's an estimate of the SEM, not the actual SEM. It uses a single sample deviation and the number of elements in that sample to make the estimate. More on reddit.com

The standard error for the mean is $\sigma \, / \, \sqrt{n}$ where $\sigma$ is the population standard deviation. So in this example we see explicitly how the standard error decreases with increasing sample size. The standard deviation is most often used to refer to the individual observations. More on stats.stackexchange.com

Standard deviation is a characteristic of a random variable describing how dispersed samples from it are, standard error (of mean) is a description of how confident you are in where the variable's mean lies. The more samples you have the smaller your standard error because you can be more confident of where the mean lies, but the underlying deviation is the same. If the standard deviation is larger, then your confidence in where the mean lies will also be broadened and you will need more samples to get to the same standard error. More on reddit.com

The variance is the average squared distance from the mean. It has lots of nice statistical properties, but it's hard to interpret because it's squared. Standard deviation is the square root of the variance. It loses a lot of the nice properties of the variance, but getting rid of the square gives a good scale for talking about how far data points are from the mean. For instance, you could discuss outliers that are more than 5 standard deviations for the mean (i.e. mean + 5 times the standard deviation), or discuss the proportion of observations that are within 1 standard deviation of the mean (in either direction). IQ tests are designed to have a mean of 100 and a standard deviation of 15 so that it's easy to know that a score of 130 is 2 standard deviations above the mean (and thus quite good!). The "standard deviation of the mean" (commonly referred to as the standard error) is what you nicely described as the "average result plus or minus error". It's on the same scale as the standard deviation (i.e. not squared), but instead of describing the variability in the individual data points, it describes the variability in the mean based on how many data points you've looked at. So with the IQ test, if you test an entire school and the average score is 103 instead of 100, the standard error is what's going to help you figure out if the school is actually above average or if the difference between 100 and 103 is just random "error". More on reddit.com