Generalized linear model is defined in terms of linear predictor
$$
\eta = \boldsymbol{X} \beta
$$
that is passed through the link function $g$:
$$ g(E(Y\,|\,\boldsymbol{X})) = \eta $$
It models the relation between the dependent variable $Y$ and independent variables $\boldsymbol{X} = X_1,X_2,\dots,X_k$. More precisely, it models a conditional expectation of $Y$ given $\boldsymbol{X}$,
$$
E(Y\,|\,\boldsymbol{X} ) = \mu = g^{-1}(\eta)
$$
so the model can be defined in probabilistic terms as
$$
Y\,|\,\boldsymbol{X} \sim f(\mu, \sigma^2)
$$
where $f$ is a probability distribution of the exponential family. So first thing to notice is that $f$ is not the distribution of $Y$, but $Y$ follows it conditionally on $\boldsymbol{X}$. The choice of this distribution depends on your knowledge (what you can assume) about the relation between $Y$ and $\boldsymbol{X}$. So anywhere you read about the distribution, what is meant is the conditional distribution.
If your outcome is continuous and unbounded, then the most "default" choice is the Gaussian distribution (a.k.a. normal distribution), i.e. the standard linear regression (unless you use other link function then the default identity link).
If you are dealing with continuous non-negative outcome, then you could consider the Gamma distribution, or Inverse Gaussian distribution.
If your outcome is discrete, or more precisely, you are dealing with counts (how many times something happen in given time interval), then the most common choice of the distribution to start with is Poisson distribution. The problem with Poisson distribution is that it is rather inflexible in the fact that it assumes that mean is equal to variance, if this assumption is not met, you may consider using quasi-Poisson family, or negative binomial distribution (see also Definition of dispersion parameter for quasipoisson family).
If your outcome is binary (zeros and ones), proportions of "successes" and "failures" (values between 0 and 1), or their counts, you can use Binomial distribution, i.e. the logistic regression model. If there is more then two categories, you would use multinomial distribution in multinomial regression.
On another hand, in practice, if you are interested in building a predictive model, you may be interested in testing few different distributions, and in the end learn that one of them gives you more accurate results then the others even if it is not the most "appropriate" in terms of theoretical considerations (e.g. in theory you should use Poisson, but in practice standard linear regression works best for your data).