What does "$\sim$" mean and $A | B \sim C$? I'm not sure if I fully understand the meaning of the symbol, I've seen this symbol in various articles but haven't managed to understand what they implied.
I did some reading and it looks like $A \sim B$ means B-Distribution of random variable $A$.
Then how would it apply in continuous case as this 
$$\epsilon_{ij}\sim F$$
$$F\sim DP$$
where $\epsilon$ is noise and DP means Dirichlet process?
Then it gets more complicated with  $A | B \sim C$ such as Dirichlet process mixture models. Like is the left-hand side denoting $\theta$ given $G$?
$$\theta_i|G \sim G$$
$$x_i|\theta_i \sim F(\theta_i)$$
source
 A: Usually ~ means "has the distribution...", so you are correct.
A ~ B, means that the random variable A has the distribution B (or is distributed equal to B).
In the article you mentioned DP is described as a distribution (see page 05, Dirichlet Processes):

A Dirichlet process DP($\alpha_0$;$G_0$) is defined to be the distribution of a random
  probability measure G over ($\theta$;$\beta$) such that...

The symbol "|" means "given" what comes next. It is used to express condition.
For example $\theta_i$ given G, has the distribution G.  
On page 7, Equation 9, the text says:

where $F(\theta_i)$ denotes the distribution of the observation $x_i$ given $i$.

A: The ~ symbol means 'distributed according to'. So $a \sim G$ means the random variable $a$ is distributed according to the distribution $G$. 
The bar symbol, as in $x_i|\theta_i \sim F(\theta_i)$, means "given". You shouldn't let it confuse you. It's often used in hierarchical models to emphasize that the lower stage of the model depends on the stage above it. 
