Hi i really need help....

when i download a new sprite or tileset to use it has a white border around it.. now i know its something to do with transparency so i tried using photoshop7 to edit it but my god! i was there for hours and didnt get no where.

So i guess im asking for a step by step guide to use photoshop7 to edit this transparency problem, I know theres easier ways of trying this and trust me ive tried 'em but they just wont work!!

If any1 has the time, then please could you tell me diferent ways of trying to sort it. Oh! im using rmxp..

Thankyou for your time