Let $X$ be a topological space. A $k$-chain in the singular chain complex of $X$ is a linear combination $c = \sum_1^k a_i\sigma_i$, where $\sigma_i$ are continuous maps from the standard $k$-simplex $\Delta^k$ to $X$. Thus we can natually define the norm of the cycle to be$$\|c\| = \sum_1^k |a_i|.$$
For $\alpha \in H_k(X, \mathbb{R})$, the Gromov norm of $\alpha$ is defined to be$$\|\alpha\| = \inf\{\|c\| : c \text{ is a singular cycle which represents }\alpha\}.$$What is the intuition behind the definition of the Gromov norm?
Part of the intuition is that if you fix the dimension $n$ and consider the set of all closed hyperbolic $n$-manifolds $X$, the Gromov norm is proportional to the volume of $X$. The proportionality constant turns out to be equal to the maximal volume of an ideal simplex in $\mathbb{H}^n$ (or the supremal volume of a finite simplex). This tells us, intuitively, that the way to acheive the infimum in the definition of $||c||$ is to choose a chain whose singular simplices lift to the universal cover $\mathbb{H}^n$ so as to be close to ideal simplices (when lifted to the universal cover).