Basically you are asking the company to do something they probably normally wouldn't do, which is to completely remove something from your credit reports from all major CRA's, in exchange for either full or partial payment. If they agree to PFD for a partial payment, make sure the account is also considered paid in full.
Always try to get this agreement in writing(or fax/email I guess), though some companies will refuse to do that, and will only agree verbally. At that point you need to decide whether to go for it & hope they keep their end of the bargain, or try again another day perhaps(maybie with a different person). Last resort you can pay it, then send Goodwill deletion letters to the company, as high up the chain as possible, & try to get the negative listing removed that way.
Depending on the listing, you may not want the whole thing removed completely, like if it's a line with late payments. You might just want the late payment listing itself removed, but keep the actual listing there. I guess in that situation it would be a GW deletion request after payment of what you owe? Can you PFD a late payment, but keep the listing? I've never had any late payments so I dunno if you can do that.
It really depends on your situation & what the listing is comprised of.
If it's medical debt, you can either attempt PFD before payment/GW delete request after payment, or you can google the Hipaa method for removal attempts(we cant discuss hipaa here).
Just remember, PFD/GW deletion attempts go against what these companies do for business, & they don't have to do this for you, so always ask nicely.
So basically all this helps you by cleaning up your reports & making you look more credit-worthy in the eyes of lenders perhaps. Your score may or may not go up depending on factors, like how many other negative listings you still have on your reports, what type of listings they are, how old the accounts being removed are, etc. It also boosts your morale seeing negative listings removed, imo. 
Just remember, some listings you will want completely removed, other listings you may want to keep, but remove any lates showing. Depends on the listing itself..all my listings were medical debt, so I had everything removed that I could.
