By the way - I have never seen the "Partial Delivery" button be active. Perhaps someone can give more info on that - is it just on wing missions for example?
"Partial delivery" is only available in very specific circumstances: if you lose part, but not all, of your mission-specific cargo.
For example: You are at Station Alpha and asked to deliver 10 tonnes of Fish to Station Beta. You pick up the 10 tonnes of mission-specific Fish from the mission board at Station Alpha - simply purchasing the Fish there from the Commodity Market is unacceptable. They need their specific Fish delivered and will not accept substitutes. You are not allowed to question why this is so - you are being paid to deliver cargo, not ask questions. So with those very particular 10 tonnes of Fish on board, you set a course for Station Beta. Unfortunately, along the way you are set upon by pirates (NPCs or CMDRs, it doesn't matter which from the point of view of this scenario). They interdict you and successfully manage to steal 2 tonnes of your precious Fish (eg. by using hatch-breaker limpets on you), and the Pirates either run away or are destroyed before you can retrieve the Fish from them. You only have 8 tonnes of mission-specific Fish left in your hold when you arrive at Station Beta.
Under the old pre-3.0 mission regime, at this point you would have been forced to Abandon that mission, cop the fine and dump the now-Stolen Fish onto the black market. Now, you can still hand in those Fish which you still have on board, using the "Partial Delivery" button. You will only get a partial reward, depending on how much of the cargo you managed to deliver. I think for less than 50% of the cargo, you can still deliver the cargo using Partial Delivery but the mission counts as "Failed" and you still get fined. For more than 50% of the Cargo, the mission is a "qualified success" and you receive a reduced reward.