This is a slang expression, which is commonly used in American English. When you take a dig at someone, you deliberately insult or annoy someone. It is also possible to say, to take a jab at someone.
Here are a few examples:
• In his speech, the Vice-Chancellor took several digs at the academic staff.
• Mala loves to take jabs at people she doesn't like.
These two expressions should be used only in informal contexts.
