GAM model summary: What is meant by "significance of smooth terms"? I have dataset for which I am constructing a GAM model, with a number of factors predicting the dependent variable. When I take a summary of the model, I get a chart that indicates the "significance of smooth terms" (which is quite significant). What does this represent?
Here is a sample of some data (totally made up btw).
gam.happiness_rating <- gam(data = ratehappiness2008, overall_happy ~ s(salary, k=3) + s(age, k=3) + as.factor(sex) + as.factor(year) + num_siblings + num_vacation)

summary(gam.happiness_rating)

Parametric coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)          68.9221     5.4937  10.432  < 2e-14 ***
as.factor(sex)1     -12.3661     3.6232  -2.55  0.02346 ** 
as.factor(year)1999  21.4689     3.3060   2.262 2.03e-06 ***
num_siblings          1.2332     0.1082   1.648  0.02235 .  
num_vacation          -4.3824   3.3261  -1.233  0.132343   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
               edf Ref.df      F  p-value    
s(salary)     2.111  1.723 15.843  < 2e-16 ***
s(age)        1.844  1.485  16.46 2.47e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

 A: As not_bonferroni mentions, help(summary.gam) do have useful information. Do see the references therein or 

Wood, Simon N.. Generalized Additive Models: An Introduction with R, Second Edition (Chapman & Hall/CRC Texts in Statistical Science).

particulier section 6.12. To give a brief and simple answer to 

When I take a summary of the model, I get a chart that indicates the "significance of smooth terms" (which is quite significant). What does this represent?

let us suppose that you only have one covariate $x_i$ and you have an outcome variable is $y_i\in\{0,1\}$ which is $1$ if observation $i$ is overall happy and $0$ otherwise. The model you fit is 
$$
g\left(E\left(y_i \mid  x_i\right)\right) = \alpha + f(x_i)
$$
where $g$ is a link function, and $f$ is an unknown smooth function. Then the $p$-value is for the null hypothesis $H_0:\, f(x_i)=0$. To give a simple example then we make some simulations below where $f(x_i)=2\sin(x_i)$, $f(x)=x$ and $f(x_i)=0$.
library(mgcv)
set.seed(2160179)
n <- 100
x <- seq(-pi, pi, length.out = n)

# f(x) = 2sin(x)
y <- 1/(1 + exp(-(1 + 2 * sin(x)))) > runif(n)
fit <- gam(y ~ s(x, k = 20), binomial())
summary(fit)
#R ...
#R Approximate significance of smooth terms:
#R        edf Ref.df Chi.sq  p-value
#R s(x) 4.285  5.344  32.61 8.33e-06 ***
#R ---
#R ...

# f(x) = x
y <- 1/(1 + exp(-(1 + x))) > runif(n)
fit <- gam(y ~ s(x, k = 20), binomial())
summary(fit)
#R ...
#R Approximate significance of smooth terms:
#R      edf Ref.df Chi.sq  p-value
#R s(x)   1      1  24.45 7.63e-07 ***
#R ---
#R ...

# f(x) = 0
y <- 1/(1 + exp(-1)) > runif(n)
fit <- gam(y ~ s(x, k = 20), binomial())
summary(fit)
#R ...
#R Approximate significance of smooth terms:
#R        edf Ref.df Chi.sq p-value
#R s(x) 6.532  8.115  11.04    0.21
#R ---
#R ...

We reject the null hypothesis in the two first cases but not in the latter as expected. Suppose now that we add two additional covaraites to the model such that
$$
g\left(E\left(y_i \mid  x_i\right)\right) = \alpha + f_1(x_{1i}) + f_2(x_{2i}) + \beta x_{3i}
$$
Your null hypothesis is that there is not a (potentially non-linear) association with covariate one, $x_{1i}$, given a (potentially non-linear) association with  covaraite two, $x_{2i}$, and a linear association with covariate three, $x_{3i}$, on the link scale.
One final comment (which is stressed in help(summary.gam)) is that the $p$-values are without considering uncertainty in the smoothing parameter estimates. Thus, you may need to be careful when the $p$-value is close to your threshold. 
A: The significance of the smooth terms is exactly what the name says: how significant the smooth terms of your model are. Perhaps the question was much more what the smooth terms are (since you seem to understand what significance is)? Your model includes various terms, some of them are "smooth" terms, basically penalized cubic regression splines. Those are the terms with an "s", i.e., s(salary, k=3) for instance. Some other terms are parametric, for instance num_siblings or num_vacation. Each of these terms is more or less important on explaining variance of your response variable "overall_happy". Some of them seem quite unimportant, like num_vacation which has a small significance (a large p value of 0.132343). This mean that this variable has probably no mechanistic or deterministic or physical influence on your response variable and, thus, you can ignore it and remove it from your model. Other terms have a high significance (a small p value), like the smooth term s(salary). This means that most probably, in reality, the salary of a person is one of the major factors contributing to its happiness. 
