Do I need to see the first Pink Panther movie to understand the second?

Assuming you are talking about the series of movies from the 1960's and 70's, then no.

I saw all the movies of the series when I was a kid, except the first (self-titled) one. I didn't see that one until I was an adult, and was actually a bit underwhelmed.

The first movie is actually a fair bit different from the rest, in that the main character of the series is just a supporting character in The Pink Panther. I'm guessing they realized Peter Sellers' character was the best thing in that movie, so they just made more movies centered around that character.

If you are instead talking about the later movies from 2006 and 2009 starring Steve Martin, I'd suggest not watching those at all. Failing that, you shouldn't watch them without watching some of the original series first, so you know what they are trying to do. But if you insist, I highly doubt there's anything going on in the second (2009) that you'd need the first (2006) to understand, so no again^*.

As a good rule of thumb, I'd suggest you don't bother with any Pink Panther movie that doesn't have Peter Sellers in it.

^{* - You will probably want to see an earlier Sellers movie in the series afterwards though, to understand why that was supposed to be funny.}