Notation Bayesian Statistics $\propto$ I often read the following notation:
$\propto$. 
How is this sign called and what is the definition of it?
 A: It means proportionallity. I'm sure that you see something like $f(\theta\mid I) \propto p(\theta)p(I\mid \theta)$.
It is a great result: posteriori density is proportional to priori density times likelihood.  Thus, it say how change our guessings about $\theta$ trought the evidence of the data.
A: $$
L(\alpha,\beta) = \text{some constant}\times x^{\alpha-1} e^{-\beta x}
$$
is written as
$$
L(\alpha,\beta) \propto x^{\alpha-1} e^{-\beta x},
$$
and here one must be aware that in this context “constant” means not depending on $\alpha$ or $\beta$.  It does not mean not depending on $x$.  That point is made clear by the left side of the relation, written as a function of $\alpha$ and $\beta$.
$$
f(x) = \text{some constant}\times x^{\alpha-1} e^{-\beta x}
$$
is written as
$$
f(x) \propto x^{\alpha-1} e^{-\beta x},
$$
and this time “constant” means not depending on $x$ but possibly depending on $\alpha$ and $\beta$.  If $f$ is supposed to be a probability density function on the interval $[0,\infty)$, then the “constant” must be the reciprocal of
$$
\int_0^\infty x^{\alpha-1} e^{-\beta x}\,dx =  \frac{\Gamma(\alpha)}{\beta^\alpha}.
$$
If $L(\alpha,\beta)$ above is supposed to be a likelihood function, then regardless of whether you're doing something like maximum likelihood estimation or multiplying the likelihood by the prior probability measure to get the posterior probability measure, knowing the value of the “constant” is of no use at all; generally one just chooses the one that's computationally least inconvenient.
