What is a random measure? What is a random measure?
The Wikipedia article is quite confusing and used to formulate only a random counting measure instead of just the random measure. I'm working on the Dirichlet process and with the more in-depth articles they start with the notion of a random measure, but there is not much intuition given. Are there perhaps multiple definitions?
From what I understand it is a mapping from a probability space $(\Omega,\mathbb{F},\mathbb{P})$ to a measurable space $(S,\Sigma)$ just as a $(S,\Sigma)$-valued random variable. 
The difference is that it is a specific $(S,\Sigma)$-valued random variable, namely one that is set of normal ($\mathbb{R}$-valued) random variables. 
Is this correct? Are there some good examples of different types of random measures?
 A: As you say, a random measure is a measure-valued RV.  You need a sigma algebra structure on the space of measures for this.  Often times, the measures will be defined on a separable metric space, and you will be able to use the Prokhorov topology.
An "equivalent" (i.e. "naturally identified") understanding of random measures in the case mentioned above is to say the following: $\alpha: \Omega\times\Sigma\rightarrow [0, 1]$ is a random measure if fixing the left coordinate always gives you a probability measure, and fixing the right coordinate always a measurable map. (See, for instance, Kechris' Classical Descriptive Set Theory Ch. 17)
Random measures are used when talking about regular conditional distributions and exchangeability.
A: There are two equivalent definitions of random measure.
Definition 1 (as a random element)
Using words:  Given a probability space, a random measure is a measure-valued random element from the sample space to a space of measures where all the measures are defined on the same $\sigma$-field of some measurable space.
Using mathematical symbols: Given a probability space $(\Omega,\mathcal{F},P)$ and a measurable space $(E,\mathcal{E})$, a random measure, $X$, is a measure-valued random element from $(\Omega,\mathcal{F},P)$ to $(\tilde{M},\tilde{\mathcal{M}})$, where $\tilde{M}$ is the space of all measures on $(E,\mathcal{E})$ and $\tilde{\mathcal{M}}$ is the $\sigma$-algebra over $\tilde{M}$.
Note: $E$ is usually taken as a separable complete metric space and $\mathcal{E}$, as the Borel $\sigma$-algebra over $E$. Further, $\tilde{M}$ is usually taken as the space of all locally finite measures $\mu$ and $\tilde{\mathcal{M}}$, the $\sigma$-algebra over such $\tilde{M}$, generated by the projection maps $\pi_B:\mu \mapsto \mu(B)$ for all $B \in \mathcal{E}$. As to why such choices are made, I quote from the footnote at P-1 of Kallenberg:

"The theory has often been developed under various metric or topological assumptions, although such a structure plays no role, except in the context of weak convergence.

Definition 2 (as a transition kernel)
Using words: Given a probability space and another measurable space, a random measure is a measurable function from the product space of the sample space and the $\sigma$-field of that measurable space to the real line, i.e., a function of two variables, such that if the first variable is fixed (to a particular value), the function becomes a particular measure on that $\sigma$-field and if the second variable is fixed (to a particular set), the function becomes a measurable function on the probability space.
Using mathematical symbols: Given a probability space $(\Omega,\mathcal{F},P)$ and a measurable space $(E,\mathcal{E})$, a random measure, $X$, is a function of two variables, taking values $\{X(\omega,B):\omega \in \Omega, B \in \mathcal{E} \}$, with $X:\Omega \times \mathcal{E} \to \mathbb{\bar{R}}$, such that for a fixed $\omega$, $X$ is a measure on $\mathcal{E}$ and for a fixed $B$, $X$ is $\mathcal{F}$-measurable. (such $X$ is, in fact, called a transition kernel from $\Omega \to E$)
Note: Same note, as above, applies.
Reference
Kallenberg -- Random Measures, Theory and Applications
