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