Dirichlet process I am studying the dirichlet process. Am I right to assume that every observation/sample from the DP, $G(T_k)$ , is a Dirichlet distribution, and that at the same time, the distribution over all cells / observations is Dirichlet as well? I am somewhat confused on the meaning of partition - and whether this is a subset of observations or a subset of the "simplex" that normally defines a Dirichlet distribution - any intuition much appreciated.

 A: A realization from a DP is a discrete distribution $G$, not a Dirichlet distribution. Basically a DP is a distribution from which you sample distributions.
If you ever studied how you can sample Bernoulli or Binomial distributions from a Beta distribution, or Multinomial from a Dirichlet distributions this should not be totally strange. Basically, since the support of a Dirichlet is the set of points $(x_1, \ldots, x_m)$ in $\mathbb{R}^m$ such that $x_i \geq 0 \ \forall i$ and $x_1 + \ldots + x_m =1$ when you get a realization $(x_1, \ldots, x_m)$ from a Dirichlet you can think of it as being a multinomial that picks sucessfully $1$ out of $m$ objects and picks the object $i$ with probability $x_i$.
A probability distribution over $\Theta$, like $G$, works also as a probability measure over $\Theta$. Roughly speaking it follows these rules:
1) attributes numbers between $0$ and $1$ to subsets of $\Theta$ (probability measures), i.e., $P(T) \in [0,1]$ for any $T \subset \Theta$.
2) $P(\cup_{i=1}^{\infty} T_i) = \cup_{i=1}^{\infty} P(T_i)$ for $T_1, T_2 \ldots$ disjoint subsets of $\Theta$
3)$P(\Theta) = 1$
So you may find $G$ being referred as a 'probabiilty measure'.
A (finite) partition $T_1, \ldots, T_K$ of $\Theta$ is a collection of disjoint subsets $T_i$ of $\Theta$ (that is, for $i \neq j$, $T_i$ and $T_j$ have no point in common), and the union of all $T_i$ is $\Theta$.
When you sample from a $DP(\alpha, H)$ you should get a discrete distribution $G$, which in turn will attribute probability mass to points of $\Theta$, and thus also to subsets of $\Theta$ (in this case integrating $G$ on a subset of $\Theta$ can be understood as a summation of the masses of probability $G$ attributes to points in this subset), and it does that in a way that for any finite partition $(T_1, \ldots , T_K)$ of $\Theta$ we have $(G(T_1), \ldots , G(T_K)) \sim Dirichlet(\alpha H(T_1), \ldots , \alpha H(T_K))$. So a realization from a DP is a discrete distribution that induces a Dirichlet distribution over partitions of $\Theta$.
Note that since $G$ is a random variable each $G(T_i)$ is also random and since $G$ is a probability measure, $G(T_i)\geq 0$, and the sum of all $G(T_i)$ is the same as $G(\Theta) = 1$.
Note too from the above that if you change your partition to another one, you'll get a different Dirichlet distribution. So a DP is a stochastic process indexed in the set of partitions of $\Theta$, rather than indexed on a subset of the real line or the positive integers as usual stochastic processes.
The scheme goes like that, from a $DP(\alpha, H)$ you sample $G$, from $G$ you can sample points $\theta \in \Theta$.
A DP is a infinite generalization of a Dirichlet distribution. So for the most part you'll find they have similar characteristics, apart from dimensionality. In both of them you sample discrete distributions from.
A: 
Am I right to assume that every observation/sample from the DP,
  () , is a Dirichlet distribution

Every sample/realization (I don't like the word observation) is a probability measure on $\Theta$. We don't assume what $\Theta$ is, so no, they don't have to be Dirichlet distributions. 
However, if you fix an arbitrary partition $T_1, \ldots, T_k$ before you sample a realization $G$, then the probabilities of those sets in the partition, $G(T_1), \ldots, G(T_k)$ follow a Dirichlet distribution. Those $k$ random probabilities are random, and because they are a probabilities and because $T_k$ is a partition (disjoint union equals the sample space), they are nonnegative and sum to $1$. 

I am somewhat confused on the meaning of partition - and whether this
  is a subset of observations or a subset of the "simplex" that normally
  defines a Dirichlet distribution - any intuition much appreciated.

$T_1 \ldots, T_k$ is a finite partition of $\Theta$ if they are all disjoint and they union to $\Theta$. It's a breakup of the entire space into nonoverlapping windows/regions. Again, I don't like the word "observations" because often these DP realizations are used as distributions for observed data (which I call observations). This is a separate issue, however.
This partition idea has nothing to do with a simplex until we start bringing $G$ realizations into the fold. We start getting a random simplex only after we start using $G$s to map regions $T_i$ to random probabilities. Even though they are random, they still are probabilities--this means that $G(T_1) + \cdots + G(T_k) = 1$. So fix a partition $T_1, \ldots, T_k$, and think of all the possible vectors $G(T_1), \ldots, G(T_k)$; then that would be your simplex, which is a subset of $\mathbb{R}^k$.
What I haven't discussed yet, is how to visualize a sample $G$. When we sample $G$ will it look like a continuous distribution (e.g. a normal density curve), or a discrete distribution (e.g. a Binomial), or something in between? According to the stick-breaking process view of a DP, these realizations $G$ are discrete with probability $1$. This is separate than talking about probabilities of partitions. Why? Well how do you calculate probabilities? Sometimes you integrate, and sometimes you sum. 
