First the algebraic definition. A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie Group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.

Then the rank of a symmetric space is the dimension of the "maximal $\mathbb{R}$-split torus", i.e. the maximal dimension of an abelian diagonalizable over $\mathbb{R}$ subgroup of $G$.

The geometric meaning is that the rank is the dimension of the maximal flat submanifold of the symmetric space. If the rank is $1$, then the maximal flats are geodesics, and the symmetric space turns out to be negatively curved.

If the rank is larger then one, then the symmetric space is only non-positively curved. However, higher-rank symmetric spaces have spectacular rigidity properties (e.g. Margulis superrigidity, arithmeticity and the normal subgroup property come to mind).

There are only three families of rank 1 symmetric spaces,

1) hyperbolic $n$-space, corresponding to the Lie group $SO(n,1)$.

2) complex hyperbolic $n$-space, corresponding to the Lie group $SU(n,1)$.

3) quaternionic hyperbolic $n$-space, corresponding to the Lie group $Sp(n,1)$.

There is also one exceptional example:

4) the Cayley upper half plane, corresponding to the Lie group $F_4^{-20}$.

The spaces 3) and 4) have some but not all of the rigidity properties of higher rank (in particular, superrigidity and arithmeticity, but not the normal subgroup property).