The notation means that you sum over all allowed $i$ and $j$ such that $i < j$. The notation is compact and protects you from errors that may come from doing the end-points incorrectly.
Here is an example. Suppose that $1\leq i \leq m$ and $1\leq j \leq n$ where $m$ and $n$ are not the same. If $m < n$, the sum becomes
$$\sum_{i < j} a_{ij} = \sum_{j = i+1}^n \sum_{i=1}^{m}a_{ij}.$$
If $n < m$ we have
$$\sum_{i < j} a_{ij} = \sum_{j = i+1}^n \sum_{i=1}^{n-1}a_{ij}.$$
Formally, these are two different expressions and to determine which one you are using can take a few seconds here but it could take a lot more time for more complicated sums.