Standard deviation (SD) quantifies variability within your data. SD does not change predictably with the sample size. Pluto uses the Bessel correction in its SD calculation.
Standard error of the mean (SEM) quantifies how far your sample mean is likely to be from the entire population mean. It takes into account the SD value and the sample size.
SEM is always smaller than SD and the larger the sample size the smaller the SEM becomes.
To show the variability within the samples from one experiment use the SD. You will need a minimum of 3 samples to calculate SD with the Bessel correction.
To show how well your multiple replicate experiments represents the mean of the entire population, use SEM. Using SEM implies that you have performed the same experiment multiple times (aka: multiple technical replicates) and for each of those you are plotting the mean for the biological replicates within each experiment.
You measure the expression of gene X in 3 control and 3 treated mice. For each group you can plot the expression level of gene X in each of the 3 mice and show the SD. This represents the variability among the 3 mice in each group.
Next you repeat this experiment in a different mouse cohort. To combine both experiments into one graph, you would calculate the mean (average) for each group in each experiment and plot that (n=2). Now you would show the SEM to represent the standard deviation among the means of the 2 experiments and how well they represent the entire population of control and treated mice.
Regardless of which calculation you choose for your error bars make sure to mention it in your graph legend.
Jaykaran (2010). "Mean ± SEM" or "Mean (SD)"?. Indian journal of pharmacology, 42(5), 329. https://doi.org/10.4103/0253-7613.70402
Nagele P. (2003). Misuse of standard error of the mean (SEM) when reporting variability of a sample. A critical evaluation of four anaesthesia journals. British journal of anaesthesia, 90(4), 514–516. https://pubmed.ncbi.nlm.nih.gov/12644429/
Evans, R. G., & Su, D. F. (2011). Data presentation and the use of statistical tests in biomedical journals: can we reach a consensus?. Clinical and experimental pharmacology & physiology, 38(5), 285–286. https://doi.org/10.1111/j.1440-1681.2011.05508.x