The direct effect means that a variable (X) does not go through (without the intervention of) a mediator (M) to produce a causal impact on the outcome (Y).  That is: 

$$X -> Y$$   ( as opposed to   $$X -> M -> Y$$)$$

Direct Effect

The indirect effect means that the variable (X) passes through the mediator (M) to cause its effect on the outcome (Y).  Said another way, the variable (X) affects the outcome (Y) by way of the mediator (M).  The indirect effect can be expressed as an amount of mediation (a measured magnitude) by the mediator (M).  

In the context of the causal diagram, the magnitude of the mediation can be observed when the variable (X) will result in the increase on the outcome (Y) while holding the variable (X) constant, while at the same time, when a unit increase in the variable (X) would achieve an increase in the mediator (M).  

Indirect Effect

The so-called "smoking gene" (rs16969968) is a genetic variant that predisposes individuals with the variant to smoking and lung cancer. This genetic variant is also known as "Mr. Big," and "The Big One." In VanderWeele's study referenced in the book, the interaction impact between the smoking gene and smoking was overlooked so the mediation analysis was inaccurate.

"Smoking Gene"

For linear causal models, total effect is the sum of direct and indirect effects, and can be calculated with the "sum of products" rule.

However, for non-linear systems, a modified version of additivity holds true:
$Total ~Effect (X = 0\rightarrow X = 1) \\= NDE (X = 0\rightarrow X = 1) – NIE (X = 1 \rightarrow X = 0)$


Total Effect

Mediation analysis is a causal analysis technique used to assess and measure the impact of different "pathways" and "mechanisms" by disentangling and differentiating between direct effect and indirect effect. As such, it serves to distinguish whether an outcome is caused directly in response to the variable of interest (the direct effect), or through mediators (the indirect effect). Assessing the counterfactuals of the situation can be a notable part of the process.

University of Michigan - Ann Arbor

We want to ask certain typical questions about a mediator: Does it account for the entire effect? Does Drug B work exclusively through blood pressure or perhaps through other mechanisms as well? 
The answer depends on whether the observed salary disparity is produced directly in response to the applicant’s sex or indirectly, through a mediator such as qualification, over which the employer has no control. All the above questions require a sensitive ability to tease apart total effects, direct effects (which do not pass through a mediator), and indirect effects (which do).

Mediation

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

In 1973, Eugene Hammel, an associate dean at the University of California, noticed a worrisome trend in the university’s admission rates for men and women. According to his data, it is shown that 44 percent of the men who applied to graduate school at Berkeley had been accepted, compared to only 35 percent of the women.
After a careful investigation, Eugene found out that even though for each apartment, the admissions decisions were consistently more favorable to women than to men, the admission rate for women is lower than that of men.
The causal story Eugene discovered is that women were rejected in greater numbers because they applied to harder departments to get into, like history and literature, therefore having a lower acceptance rate.

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