Can an irrelevant variable be significant in a regression model? I'm reflecting about statistical significance and I think a variable can be irrelevant and significant at the same time. But, how do I explain that? And if that were the case, should I include that variable to my regression model?
 A: "Relevant" and "irrelevant" are not well-defined statistical terms that everyone agrees on.
Case in point: the answer by @Demetri Pananos interprets "relevant" as "causal" and the answer by @Sextus Empiricus interprets "relevant" as "included in the model that we assume is the true — or at least a valid — model for the data". Both interpretations are meaningful but they are not equivalent.
Consider for example a (correctly specified) model for the response to a medical treatment vs placebo that includes gender and age effects: $E(Y) = \beta_a\textrm{age} + \beta_g\textrm{gender} + \beta_t\textrm{treatment}$. The effect of the treatment, $\beta_t$ is causal: it's the difference between the outcome if a patient is given the treatment vs if the patient is given a placebo. However the patient doesn't "cause" the outcome of his/her/their own treatment with personal biological and genetic characteristics. A better term for age and gender in this model is "covariates" or "effect modifiers". They are certainly relevant in order to prescribe each patient the most appropriate treatment.
Another popular attempt to specify what "relevant" means is to make a distinction between "practically significant" and "clinically significant" on one hand and "statistically significant" on the other hand. For example, an effect size might be statistically different from 0 but very small in magnitude  (absolute value), so perhaps not particularly useful in understanding phenomenon X, the argument goes.
Practical vs Statistical significance 
Is there a colloquial way of saying "small but significant"? 
There are subtleties with this interpretation of relevance has as well:
Stop talking about “statistical significance and practical significance”
So one takeaway is: If someone uses the term "(ir)relevant", "(in)significant", etc. to discuss their analysis, ask them "What exactly do you mean when you say your result is [a generic term hinting at importance]?"
A: 
I think a variable can be irrelevant and significant at the same time. But, how do I explain that?

This can be explained by using the concept of type I errors.
Below is an example by repeating a t-test 1000 times where we test whether the random number generator has a mean different from zero. Say that we consider a p-value below 0.05 significant, the we find 41 times that the mean of the number generator is significantly different from zero.

p = rep(0,1000)
for (i in 1:1000) {
   set.seed(i) 
   p[i] = t.test(rnorm(100,0,1))$p.value
}

plot(p, bg = 1 + (p<0.05), col = 1 + (p<0.05), pch = 21, cex = 0.5,
     main = "p values in 1000 simulations of a t-test with different seed\n 41 cases of significant p<0.05 level", xlab = "seed number", ylab = "p-value")
lines(c(0,1000),c(0.05,0.05), lty =2)

Significant observations may happen even when there is no true effect (the null hypothesis is true) because sampling is subject to random statistical variations.
The point of 'significance' is to tell whether some observation is extreme and express it in terms of a probability, but an extreme/significant observation does not mean that it is not able to occur when there is not a true (relevant) effect. An extreme/significant observation can happen by chance. (Like in the figure above it happened 41 times out of 1000 experiments)
So that is what significant means from a statistical point of view

an extreme observation that falls outside the range of likely statistical variations that might be expected.

Significance does not directly mean a 'relevant' variable.
This is a bit of a language problem as well. Statistical significance relates to probability of observations. It is not 'significance' as in 'important' as it is used in common language.
Significance is also mostly important when it is absent. Significance is more like a minimal condition for some observation/variable to be important, and it is not like a sufficient condition.
A: Complementary to Demetri's nice answer (+1):
Aside (unmeasured) confounding, we might have spurious associations that are not the same an unmeasured confounding. Two variables X and Y might be independent but have with similar covariance matrices that looks indistinguishable to confounding due to network or spatial structures. A reasonable fresh reference on the matter is: Network Dependence Can Lead to Spurious Associations and Invalid Inference (2020) by Lee & Ogborn where the authors refer to this phenomenon as "spurious associations due to dependence".
A: Here is a good example.  Suppose you are interested in modelling the effect of ice cream sales on incidence of shark attacks.  Now, clearly there is no association; buying ice cream in no way affects the incidence of shark attacks.  However there is a third variable which affects both, namely the temperature outside.
On hot days, people will want ice cream and might also want to go swimming.  Hence, hot days see increases in both ice cream sales and shark attacks (by virtue of more people going swimming and hence being at risk for an attack). This is known as confounding.
So clearly, ice cream sales is irrelevant when studying shark attacks but were we to regress shark attack numbers on ice cream sales we would find a significant result.  However, that statistical significance is confounded by temperature, and so the result is meaningless.
EDIT: Because this has generated conversation around the interpretation of "irrelevant" I feel the need to make some additional comments and concede I should have said "it depends" (even though I object to the arguments made).  If "relevant" is understood to mean "closely connected or appropriate to what is being done or considered" then "irrelevant" should be taken to mean "not closely connected or appropriate...".  In this case, ice cream sales could be considered "relevant" since they would be correlated -- but I find interpreting "relevant" in a purely statistical sense to be an extremely narrow (and irregular) way to attach meaning to "relevant".  None the less, it is an interpretation one may have.
If, however, you interpret relevance or "closely connected..." in a mechanistic sense -- i.e. how closely connected are ice cream sales and shark attacks in the sense that I could change the former and affect the latter or vice versa -- then ice cream sales would be considered irrelevant and my original comment applies.
A: The definition of significance is that there is a certain probablility (not more than $\alpha=0.05$) that the wrong conclusion was drawn. The result of a statistical test can be significant, i.e. it can state that whatever the respective test tests for holds with high probability.
For example, the test result can state that there is a high probability that the weight of individual apples follows a Gaussian distribution. Or that there is a high probability that the weight of individual apples does not follow a Gaussian distribution. (Note that one of those two cases is wrong, i.e. the test yielded a wrong result. The probability for a wrong but significant result is not more than $\alpha=0.05$.) Or the test result might state that there is a high probability that ice cream consumption and shark attack frequency are correlated. Or that there is a high probability that ice cream consumption does not cause shark attacks. Or that there is a high probability that shark attacks do not cause higher ice cream consumption. Depends on which statistical test is applied to what data.
However, I don't know what you mean by a variable being "significant". The result of a statistical test can be significant. One variable can play different roles in statistical tests.
