Items with "Unique" listed in the discription do not stack the effects that follow the "Unique" word.
Take BV for example:
+375 Health +375 Mana +50 Magic Resist UNIQUE Passive: Blocks one enemy ability every 45 seconds.
So if you were to buy 2 banshee veils you would get the +mr + health + mana but you would not get 2 shields around you, instead you would only get the effect of one of those items.
Stacking items does not have diminishing returns in the sense that the second item you add provides the same stat boosting benefits as the first, but it can provide diminishing returns in the overall value that stat...
Take for example BV:
One item would give you +375 Health +375 Mana +50 Magic Resist
Two items will give you +750 Health +750 Mana +100 Magic Resist (+375 Health +375 Mana +50 Magic Resist for each one)
So the second item still gives the same stat boosts as the first, however armor/mr does have diminishing returns, which means at a certain point it is more benificial to start building health for survivability than additional armor/mr.
AD/AP also are effected by diminishing returns in that at a certain point increasing your armor/magic pen provides more damage output than simply increasing your AD/AP.
One item that some people do stack even though it has a unique ability is Archangels Staff, because it can give you so much AP. It is debatable if AA stacking is even a good option because it is quite expensive and you lose out on other items which could give you more ap damage by increasing your magic pen instead of simply adding ap (again back to the diminishing returns).
Some people will stack Rod of Ages, Bloodthirsters, phantom dancers, hextech gunblade. Typically people who stack will stack items that provide a significant offensive bonus and they do this because they have moves that scale particularly well for a stat.
These Items I have just listed would be considered big item purchases and are much harder to stack. People often times will stack very cheap items, most commonly seen is "doran stacking." Some ap champs may stack multiple dorans rings, ad champs multiple dorans blades, you rarely see doran shield stacking (again usually looking for offensive bonuses from stacking).
Dorans stacking provides sustainability in lane either through mana regen or life steal, fair amount of damage, with some additional health. Overall it is a very cost efficient item which is why it can be stacked. The major disadvantage with Doran stacking is that they do not build into anything else, so ultimately you will sell them at a loss or hang on to them throughout the game, which can make it take longer to afford those bigger items.
The goal with dorans stacking is to have such a successful laning phase you more than make up for the loss in gold, from a combination of CS/champion kills, or having such a lane dominance you starved your opponent of those things.