physicist call

[Kamog]

is that that thing when you drop an apple and a feather and they fall at the same rate?

Yes, but only in a vacuum. When you drop them in the atmosphere, the feather falls slower because its shape has greater air resistance.

or is it more like when you get stuck to the core of jupiter and can't get up?

The surface gravity of Jupiter is 2.53G. You would weight 2.53 times as much on Jupiter.

*Theoretically can gravity fields be used to stop particles, no matter the size/type/velocity? (shields stopping APBs, PPBs, DUCs and so on)

Yes, but you will need an extremely powerful gravitational field.

*What are gravitons? How are they related to gravity? (same root)

A graviton is a theoretical particle that carries the gravitational force. Its existence has not yet been confirmed experimentally.

Question: is "annihilation" of matter in large quantities somehow related to gravity?

Yes, not only does matter annihilation produce a large amount of energy, the reaction will also produce gravity waves. Gravity waves could possibly be used as a weapon if they could be focused and directed.