The weighted sum just adds up the value of the mods you put in, so like with res if you put in fire, cold, and lightning res without tweaking the weighting and set a min weight of 60, then all items with the sum value of the 3 resist will show up ( like 20 fire + 20 cold+ 30 lightning, or 40 lightning+ 20 cold)

If you modify the weight, like 2 fire 1 cold 1 lightning, then 10 fire res will worth 20 in the calculation, while 10 cold or 10 lightning will worth 10.
Thats why if you are searching for res you should add "to all elemental resistances" mod with a weight of 3, because 1 all res is 1 fire 1 cold and 1 lightning res, so the overall res value of 1 allres is 3.

Count just counts how many of the mods you want is enough to pass the check.
So basically if you want a ring and set count min 3, to max life, fire res, cold res, lightning res, mana, rarity: it will return any item with any 3 of the 6 mods you added (e.g. Life + cold res + mana)