The coefficient of an explanatory variable in a multiple regression tells us the relationship of that explanatory variable with the dependent variable. All this, while 'controlling' for the other explanatory variables.
How I have viewed it so far:
While each coefficient is being calculated, the other variables are not taken into account, so I consider them to be ignored.
So am I right when I think that the terms 'controlled' and 'ignored' can be used interchangeably?
Controlling for something and ignoring something are not the same thing. Let's consider a universe in which only 3 variables exist: $Y$, $X_1$, and $X_2$. We want to build a regression model that predicts $Y$, and we are especially interested in its relationship with $X_1$. There are two basic possibilities.
we could assess the relationship between $X_1$ and $Y$ while ignoring $X_2$:
$$ Y = \beta_0 + \beta_1X_1 $$
Granted, these are very simple models, but they constitute different ways of looking at how the relationship between $X_1$ and $Y$ manifests. Often, the estimated $\hat\beta_1$s might be similar in both models, but they can be quite different. What is most important in determining how different they are is the relationship (or lack thereof) between $X_1$ and $X_2$. Consider this figure:
In this scenario, $X_1$ is correlated with $X_2$. Since the plot is two-dimensional, it sort of ignores $X_2$ (perhaps ironically), so I have indicated the values of $X_2$ for each point with distinct symbols and colors (the pseudo-3D plot below provides another way to try to display the structure of the data). If we fit a regression model that ignored $X_2$, we would get the solid black regression line. If we fit a model that controlled for $X_2$, we would get a regression plane, which is again hard to plot, so I have plotted three slices through that plane where $X_2=1$, $X_2=2$, and $X_2=3$. Thus, we have the lines that show the relationship between $X_1$ and $Y$ that hold when we control for $X_2$. Of note, we see that controlling for $X_2$ does not yield a single line, but a set of lines.
Another way to think about the distinction between ignoring and controlling for another variable, is to consider the distinction between a marginal distribution and a conditional distribution. Consider this figure:
(This is taken from my answer here: What is the intuition behind conditional Gaussian distributions?)
If you look at the normal curve drawn to the left of the main figure, that is the marginal distribution of $Y$. It is the distribution of $Y$ if we ignore its relationship with $X$. Within the main figure, there are two normal curves representing conditional distributions of $Y$ when $X_1 = 25$ and $X_1 = 45$. The conditional distributions control for the level of $X_1$, whereas the marginal distribution ignores it.
They are not ignored. If they were 'ignored' they would not be in the model. The estimate of the explanatory variable of interest is conditional on the other variables. The estimate is formed "in the context of" or "allowing for the impact of" the other variables in the model.
