How will changing the units of explanatory variables affect a regression model? Let's say I am predicting weight from height (cm) and age (years). Then I decided to convert height into meters and age into months. The interpretation of the coefficients out of the regression might be not that straightforward (I will need to think in meters and months) but I will still get my accurate regression, won't I?
I also thought that the by converting the age into months might be bad for the prediction, as people's (grown-ups) weight do not change drastically within the months, it might change drastically within the years. So, I will get high variance and I might overfit, is that correct?
 A: Nature is indifferent to units of measurement.  Thus there will be no untoward effects of changing the units of your explanatory variables in a regression model.  
If people's weight increases over years, then it will increase, on average, 1/12 as much over months.  Thus, the estimated slope in the model using months will be precisely 1/12 the estimated slope in the model using years.  Likewise, if people who are taller in centimeters tend to be heavier, then those same people, when measured in meters, will also tend to have the same association between their height and weight, but the estimated slope will be 100 times larger.  
Changes of units will not affect the goodness of fit of your model, its out of sample predictive accuracy, nor will it make you more or less likely to overfit.  
A: As @shf8888's comment notes, @gung's answer only applies to (simple) linear regression. In principle the units should not matter, but in other regression models this may not be automatic or easy to accomplish. 
For instance, consider a simple 2D kernel regression, where 
$$ \hat{E}(Y | X = x) = \frac{\sum_{i=1}^n K_h(x - x_i)y_i}{\sum_{i=1}^n K_h(x - x_i)}$$
and
$$K_h(x - x_i) = K\left( \frac{x - x_i}{h} \right)$$
Here, if you change the units of $x$, the regression output will only stay the same if you also change the bandwidth parameter $h$ accordingly.
In this example it is fairly easy to keep the results the same under unit changes, but if you imagine the higher-dimensional version of this model then it's easy to see how it could become difficult to adjust.
In more complex models, it may be so difficult to maintain this equivalence that the choice of units actually becomes an important part of model tuning and analysis.
A: Shifting the regressor will not change the slope but the intercept.
$Y_{i} = \beta_{0} + \beta_{1}X + \epsilon_{i} = \beta_{0} + a\beta_{1} + \beta_{1}(X-a) + \epsilon_{i} = (\beta_{0} + a\beta_{1}) + \beta_{1}(X-a) + \epsilon_{i} = \beta^{*}_{0} + \beta_{1}(X-a) + \epsilon_{1}$
ie; shifting the reggression (for example height) will change the intercept only
Scaling the regressor will change the slope but not the intercept.
$Y_{i} = \beta_{0} + \beta_{1}X + \epsilon_{i} = \beta_{0} + (\beta_{1}/a) (X*a) + \epsilon_{i}$
