What is a suppressor variable in multiple regression and what might be the ways to display suppression effect visually (its mechanics or its evidence in results)? I'd like to invite everybody who has a thought, to share.
There exist a number of frequenly mentioned regressional effects which conceptually are different but share much in common when seen purely statistically (see e.g. this paper "Equivalence of the Mediation, Confounding and Suppression
Effect" by David MacKinnon et al., or Wikipedia articles):

Mediator: IV which conveys effect (totally of partly) of another IV
to the DV.
Confounder: IV which constitutes or precludes, totally or
partly, effect of another IV to the DV.
Moderator: IV which, varying,
manages the strength of the effect of another IV on the DV.
Statistically, it is known as interaction between the two IVs.
Suppressor: IV (a mediator or a moderator conceptually) which inclusion
strengthens the effect of another IV on the DV.

I'm not going to discuss to what extent some or all of them are technically similar (for that, read the paper linked above). My aim is to try to show graphically what suppressor is. The above definition that "suppressor is a variable which inclusion strengthens the effect of another IV on the DV" seems to me potentially broad because it does not tell anything about mechanisms of such enhancement. Below I'm discussing one mechanism - the only one I consider to be suppression. If there are other mechanisms as well (as for right now, I haven't tried to meditate of any such other) then either the above "broad" definition should be considered imprecise or my definition of suppression should be considered too narrow.
Definition (in my understanding)
Suppressor is the independent variable which, when added to the model, raises observed R-square mostly due to its accounting for the residuals left by the model without it, and not due to its own association with the DV (which is comparatively weak). We know that the increase in R-square in response to adding a IV is the squared part correlation of that IV in that new model. This way, if the part correlation of the IV with the DV is greater (by absolute value) than the zero-order $r$ between them, that IV is a suppressor.
So, a suppressor mostly "suppresses" the error of the reduced model, being weak as a predictor itself. The error term is the complement to the prediction. The prediction is "projected on" or "shared between" the IVs (regression coefficients), and so is the error term ("complements" to the coefficients). The suppressor suppresses such error components unevenly: greater for some IVs, lesser for other IVs. For those IVs "whose" such components it suppresses greatly it lends considerable facilitating aid by actually raising their regression coefficients.
Not strong suppressing effects occurs often and wildly (an example on this site). Strong suppression is typically introduced consciously. A researcher seeks for a characteristic which must correlate with the DV as weak as possible and at the same time would correlate with something in the IV of interest which is considered irrelevant, prediction-void, in respect to the DV. He enters it to the model and gets considerable increase in that IV's predictive power. The suppressor's coefficient is typically not interpreted.
I could summarize my definition as follows [up on @Jake's answer and @gung's comments]:

Formal (statistical) definition: suppressor is IV with part
correlation larger than zero-order correlation (with the dependent).
Conceptual (practical) definition: the above formal definition + the zero-order
correlation is small, so that the suppressor is not a sound predictor
itself.

"Suppessor" is a role of a IV in a specific model only, not the characteristic of the separate variable. When other IVs are added or removed, the suppressor can suddenly stop suppressing or resume suppressing or change the focus of its suppressing activity.
Normal regression situation
The first picture below shows a typical regression with two predictors (we'll speak of linear regression). The picture is copied from here where it is explained in more details. In short, moderately correlated (= having acute angle between them) predictors $X_1$ and $X_2$ span 2-dimesional space "plane X". The dependent variable $Y$ is projected onto it orthogonally, leaving the predicted variable $Y'$ and the residuals with st. deviation equal to the length of $e$. R-square of the regression is the angle between $Y$ and $Y'$, and the two regression coefficients are directly related to the skew coordinates $b_1$ and $b_2$, respectively. This situation I've called normal or typical because both $X_1$ and $X_2$ correlate with $Y$ (oblique angle exists between each of the independents and the dependent) and the predictors compete for the prediction because they are correlated.

Suppression situation
It is shown on the next picture. This one is like the previous; however $Y$ vector now directs somewhat away from the viewer and $X_2$ changed its direction considerably. $X_2$ acts as a suppressor. Note first of all that it hardly correlates with $Y$. Hence it cannot be a valuable predictor itself. Second. Imagine $X_2$ is absent and you predict only by $X_1$; the prediction of this one-variable regression is depicted as $Y^*$ red vector, the error as $e^*$ vector, and the coefficient is given by $b^*$ coordinate (which is the endpoint of $Y^*$).

Now bring yourself back to the full model and notice that $X_2$ is fairly correlated with $e^*$. Thus, $X_2$ when introduced in the model, can explain a considerable portion of that error of the reduced model, cutting down $e^*$ to $e$. This constellation: (1) $X_2$ is not a rival to $X_1$ as a predictor; and (2) $X_2$ is a dustman to pick up unpredictedness left by $X_1$, - makes $X_2$ a suppressor. As a result of its effect, predictive strength of $X_1$ has grown to some extent: $b_1$ is larger than $b^*$.
Well, why is $X_2$ called a suppressor to $X_1$ and how can it reinforce it when "suppressing" it? Look at the next picture.

It is exactly the same as the previous. Think again of the model with the single predictor $X_1$. This predictor could of course be decomposed in two parts or components (shown in grey): the part which is "responsible" for prediction of $Y$ (and thus coinciding with that vector) and the part which is "responsible" for the unpredictedness (and thus parallel to $e^*$). It is this second part of $X_1$ - the part irrelevant to $Y$ - is suppressed by $X_2$ when that suppressor is added to the model. The irrelevant part is suppressed and thus, given that the suppressor doesn't itself predict $Y$ any much, the relevant part looks stronger. A suppressor is not a predictor but rather a facilitator for another/other predictor/s. Because it competes with what impedes them to predict.
Sign of the suppressor's regression coefficient
It is the sign of the correlation between the suppressor and the error variable $e^*$ left by the reduced (without-the-suppressor) model. In the depiction above, it is positive. In other settings (for example, revert the direction of $X_2$) it could be negative.
Suppression example
Example data:
         y         x1         x2

1.64454000  .35118800 1.06384500
1.78520400  .20000000 -1.2031500
-1.3635700 -.96106900 -.46651400
 .31454900  .80000000 1.17505400
 .31795500  .85859700 -.10061200
 .97009700 1.00000000 1.43890400
 .66438800  .29267000 1.20404800
-.87025200 -1.8901800 -.99385700
1.96219200 -.27535200 -.58754000
1.03638100 -.24644800 -.11083400
 .00741500 1.44742200 -.06923400
1.63435300  .46709500  .96537000
 .21981300  .34809500  .55326800
-.28577400  .16670800  .35862100
1.49875800 -1.1375700 -2.8797100
1.67153800  .39603400 -.81070800
1.46203600 1.40152200 -.05767700
-.56326600 -.74452200  .90471600
 .29787400 -.92970900  .56189800
-1.5489800 -.83829500 -1.2610800

Linear regression results:

Observe that $X_2$ served as suppressor. Its zero-order correlation with $Y$ is practically zero but its part correlation is much larger by magnitude, $-.224$. It strengthened to some extent the predictive force of $X_1$ (from r $.419$, a would-be beta in simple regression with it, to beta $.538$ in the multiple regression).
According to the formal definition, $X_1$ appeared a suppressor too, because its part correlation is greater than its zero-order correlation. But that is because we have only two IV in the simple example. Conceptually, $X_1$ isn't a suppressor because its $r$ with $Y$ is not about $0$.
By way, sum of squared part correlations exceeded R-square: .4750^2+(-.2241)^2 = .2758 > .2256, which would not occur in normal regressional situation (see the Venn diagram below).
Suppression and coefficient's sign change
Adding a variable that will serve a supressor may as well as may not change the sign of some other variables' coefficients. "Suppression" and "change sign" effects are not the same thing. Moreover, I believe that a suppressor can never change sign of those predictors whom they serve suppressor. (It would be a shocking discovery to add the suppressor on purpose to facilitate a variable and then to find it having become indeed stronger but in the opposite direction! I'd be thankful if somebody could show me it is possible.)
Suppression and coefficient strengthening
To cite an earlier passage: "For those IVs "whose" such components [error components] it suppresses greatly the suppressor lends considerable facilitating aid by actually raising their regression coefficients". Indeed, in our Example above, $X_2$, the suppressor, raised the coefficient for $X_1$. Such enhancement of the unique predictive power of another regressor is often the aim of a suppressor to a model but it is not the definition of suppressor or of suppression effect. For, the aforementioned enhancement of another predictor's capacity via adding more regressors can easily occure in a normal regressional situation without those regressors being suppressors. Here is an example.
   y       x1       x2       x3

   1        1        1        1
   3        2        2        6
   2        3        3        5
   3        2        4        2
   4        3        5        9
   3        4        4        2
   2        5        3        3
   3        6        4        4
   4        7        5        5
   5        6        6        6
   4        5        7        5
   3        4        5        5
   4        5        3        5
   5        6        4        6
   6        7        5        4
   5        8        6        6
   4        2        7        7
   5        3        8        8
   6        4        9        4
   5        5        3        3
   4        6        4        2
   3        2        1        1
   4        3        5        4
   5        4        6        5
   6        9        5        4
   5        8        3        3
   3        5        5        2
   2        6        6        1
   3        7        7        5
   5        8        8        8

Regressions results without and with $X_3$:

Inclusion of $X_3$ in the model raised the beta of $X_1$ from $.381$ to $.399$ (and its corresponding partial correlation with $Y$ from $.420$ to $.451$). Still, we find no suppressor in the model. $X_3$'s part correlation ($.229$) is not greater than its zero-order correlation ($.427$). Same is for the other regressors. "Facilitation" effect was there, but not due to "suppression" effect. Definition of a suppessor is different from just strenghtening/facilitation; and it is about picking up mostly errors, due to which the part correlation exceeds the zero-order one.
Suppression and Venn diagram
Normal regressional situation is often explained with the help of Venn diagram.

A+B+C+D = 1, all $Y$ variability. B+C+D area is the variability accounted by the two IV ($X_1$ and $X_2$), the R-square; the remaining area A is the error variability. B+C = $r_{YX_1}^2$; D+C = $r_{YX_2}^2$, Pearson zero-order correlations. B and D are the squared part (semipartial) correlations: B = $r_{Y(X_1.X_2)}^2$; D = $r_{Y(X_2.X_1)}^2$. B/(A+B) = $r_{YX_1.X_2}^2$ and D/(A+D) = $r_{YX_2.X_1}^2$ are the squared partial correlations which have the same basic meaning as the standardized regression coefficients betas.
According to the above definition (which I stick to) that a suppressor is the IV with part correlation greater than zero-order correlation, $X_2$ is the suppressor if D area > D+C area. That cannot be displayed on Venn diagram. (It would imply that C from the view of $X_2$ is not "here" and is not the same entity than C from the view of $X_1$. One must invent perhaps something like multilayered Venn diagram to wriggle oneself to show it.)

P.S. Upon finishing my answer I found this answer (by @gung) with a nice simple (schematic) diagram, which seems to be in agreement with what I showed above by vectors.
