Like a fancy thing, in many academic papers, people says 'having adjusted for age (or sex), we found that.....'
That 'adjustment' bases on a very key concept in statistics, confounder.
Let's say you are explaining something, using A as an explanatory (or determining) factor, which explains changes in B (dependent or outcome) variable.
Let's say A = age, B = number of cigars one people smoke per day.
Let's say you are interested in know whether changes in A (age) would lead a change in B (Change in B is called 'effect', or the number of cigar one smokes per day)
Now, according to CDC, confounders are "Factors that distort or mask the true effect of exposure in an epidemiologic study".
In above example, such factors could be sex (men are known to smoke more than women), drinking pattern (people who drink are more likely to smoke).
Now, what to do? How to deal with these?
The most common way is to do a stratified analysis. In this analysis, you divide your sample into groups with different possible confounding factors (i.e: male/female, in this case it is called 'adjusting for sex). You compare mean of cigars consumed per day per person in men group with that mean in women group. See if there is any difference between them, if yes, see if that is a significant difference, using p value. If P < 0.05 (the possibility that the difference has happened by chance is only 5%), then the difference is real. Sex is really a confounder.
If the dependent variable is not a quantitative one (like number of cigars we have seen), but a categorical one (like a yes/no, disease/no disease variable), then the best measure to use is odds ratio.
When the data is large, with many potential confounding factors, we have to calculate a lot of odds ratios (each is for one confounding factor). In that case, data are divided into many smaller strata. That would be time to use The Mantel-Haenszel Test, which can produce one odds ratio for all strata you have created.
See this website for epidemiological definitions.
www.cdc.gov/niosh/2001-133o.html
