Is there a combinatorial interpretation of the triangular numbers?

Solution 1:

Imagine a row of $n{+}1$ buttons underneath the triangle (as an extra row). Then for any two of those buttons you select they will designate a point of the triangle, and every point of the triangle can be identified with a pair of buttons:

Edit: David K notes in comments that a route distance triangle is a practical application of this idea. Pick two locations, read off the distance at the intersecting point of the triangle. Adapted slightly from the image given:

Solution 2:

Consider a set of $n$ people, each shaking hands with one another. How many handshakes are there? There are $\binom{n}{2}$ pairs of people, so there are $\binom{n}{2}$ handshakes.

Now imagine person $1$ goes down the line of other people and shakes hands with everyone. Then person $2$ goes down the line, shaking hands with everyone but person $1$ (since they've already shaken hands). Repeat until every person has shaken hands with every other.

Then person $1$ shook hands with $n$ people, person $2$ shook hands with $n - 1$ people, and so on, for a total of $\sum_{k=1}^{n-1}k$ handshakes.

Thus, $\binom{n}{2} = \sum_{k=1}^{n-1}k$.