could you please see which one is correct?
1. Today, on the 16th of august, there was a big earthquake....
2. Today, the 16th of august, there was a big earthquake..
Both are acceptable. "on" is often omitted in that context. Moreover, the second one looks like telegraph speech: the way in which newspaper headlines are written.
If I had to choose, I'd say, the first one is better.
Note, months of the year are capitalized; e.g., August.