In the Hamiltonian theory of gravity, see for example comment after equation (5) in "Quantum gravity with a positive cosmological constant", Lee Smolin or end of page 13 in "Background Independent Quantum Gravity: A Status Report" Ashtekar, Lewandowski.

the conjugate momenta to the connection is a vector density of weight 1. I don't know why it is a 'vector density of weight 1'.

The definition that I am aware of density is:

given a vector space $V$ of dimension $n$, a $k$-density on $V$ is a function $\mu$ from $\wedge^n V^*\setminus \{0\}$ to the reals such that

$\mu (t\alpha) = |t|^k \mu(\alpha)$, $\alpha \in \wedge^n V^*\setminus \{0\}$

Is this what should I understand as vector density?