These are the standard summoners for most ADC's (Attack Damage Carries). Ignite gives you a strong early fight which is a hard time for most MF (Miss Fortune) players because she scales very well into late-game. That is the time when you want to have just a little bit of extra dmg so you wont get set back too much. Ignite also allows you to fight at level 1 vs an enemy who has heal because Ignite does not only deal dmg over time but also cuts all the healing the target receives by half. This is not only their own healing abilities but also from other players.
Flash is a very strong Summoner Spell which allows you to relocate to the direction of your mouse cursor. Flash allows you to move around in teamfights very effectively and also lets you dodge skillshots which could otherwise cause a lot of trouble. As ADC you are the highest or second highest dmg dealer in the team so you want to stay alive and keep dealing dmg. That is why you need Flash, to stay alive and possibly get some kills to get your dmg-output just a little bit higher

These are the summoners for the "defensive" Miss Fortune build. Cleanse allows you to have strong teamfights where getting focused is not as big as a problem as when you are running ignite. Cleanse can sometimes keep you alive longer so you will deal more dmg which can win teamfights and sometimes win the game so very important! Also because of your W you dont really need ignite as your base dmg is already very high and the grievous wounds will already be applied. Flash is standard summoner on adc so no further explaination is needed

How about a little dmg? so lets start with a Longsword allowing you to farm easier and to bully your lane in combination with your Ignite, if you are lucky, you might just get an early kill. If MF gets an early kill, you can easily snowball(snowballing is the acting like a snowball, as you roll down the hill you gain speed and volume. Translated to LoL this means you keep getting stronger at the cost of your opponents) into lategame. So when you get back it's wise to pick up your Tier 1 boots and another Longsword to make sure your dmg-output stays high in the early game. The buy you get after this is 1 of 2 choices usually (unless you get some kills and you have a lot of money) you can buy a brutalizer to get a lot of dmg making it easier for your team to contest dragon or let you roam and try to pick up kills in other lanes. The other option is a vampiric sceptre. This item lets you stay in lane longer so you can keep farming and if you want to trade with your opponent, you can use your lifesteal to get your health back up if you want to. After you can finish both your BT (Bloodthirster) and your BC (Black Cleaver), again in the order you think is best for the moment. When you get the dmg you can start to focus on attack speed and critical strike chance. The best item you can get after is a Statikk Shiv because that in combination with your W gives you a lot of attack speed letting you deal massive dmg at a high speed. And if you have not melted the enemy turrets and champs at this moment of the game, feel free to buy an Infinity Edge to maximize the dmg-output and to let about half the attacks deal 2.5 times your normal dmg which stacks nice with your high attack speed and your BC-passive. For a stronger Teamfight you can choose to buy a Guardian angel to let you come back from the death to just put out a bit more dmg and letting you be just a bit tankier.
A good alternative for Statikk Shiv if you think you can farm very well without it is a PD (Phantom Dancer) this items gives you greater dmg-output at the cost of the lane clearing ability.

I agree, at first these items seem quite unconventional but the Cloth start allows you to stay in lane much longer and also because you can just use your Q as a usefull source of dmg you can still win trades despite opponents going longswords or dorans. When you can already win trades you have to make sure you dont lose when the enemy comes back i advise to get double longsword when you go back together with some more pots if needed. then you can choose if you first want to build your brutalizer which is very nice when playing against a no-sustain lane or you can go vamp sceptre and again rely on sustain and when you dont want to really fight your enemy which can be caused by you having go up against your counter (like Ashe or Draven). Then you can finish your nice ninja tabi and have a bit stronger lane again. After you can finish your bloodthirster and Black Cleaver again in the order you feel like is best. Usually this is Black Cleaver before Bloodthirster because of the huge amount of extra dmg. When you then get a Zeal or a complete Phantom Dancer you are ready for teamfights and people will have to be scared when facing you. Then (because this is your "defensive" build) you go Guardian angel to let you stay in fights for jsut a bit longer and complete the build with the Mercurial Scimitar which is the item to let you keep your dmg-output lategame and also allows you to fight AP-nukes because they wont be able to Nuke you down in one go when your Guardian Angel is down.

The Pros and Cons of the Offensive Build:

Pros:

• The dmg will continue to stay unbelievebly high throughout the game

• Whatever the enemy adc builds, you will always deal the same if not more dmg

• Dmg counters a lot of tankiness, the only thing which can actually counter you late-game is very hard engage (like flash into Unstoppable Force Malphite) and Thornmail combined with high DMG (like Olaf going in with Ragnarok and Undertow)

• Your ultimate can melt a team when used correctly (like in combination with Amumu, Sona or Zyra ultimate)

Cons:

• When you get engaged uppon, you will almost certainly die and as you are usually the focus of the enemy team you need very good map awareness or/and very good positioning

• Getting hit by even one auto-attack from a support drastically drops your movement speed allowing the other team to engage on you even easier and again, you will drop very easy from the focus you will get.

• You can get killed while chasing, when the enemy is tanky enough and you decide to chase you can get dropped very fast if the opponent wants to turn around.

The Pros and Cons of the Defensive Build:

Pros:

• A huge pro is that you wont get instagibbed in a teamfight when you are getting caught out.

• You can 1 vs 1 against bruisers easier in the mid and early lategame because you are a lot tankier yourself the bruiser will actually work as a bruiser and unlike normal vs a ADC or APC they will not drop you within seconds.

• You can do all-in in teamfights because you just wont drop that easy you can be a bit more reckless and stay just stay a bit longer before dieing or having to flee.

• When you get a strong support like Nunu you can still deal enough dmg instead of falling off compared to other, full AD ADC's.

Cons:

• You just have less dmg than other ADC's which can affect the game when it goes very late

• When items and Summoners are down you are an easy target because you wont deal a lot of dmg so it is easy to pick you off and you are not so tanky that you can just run it off.

• If your lane gets swapped early and you are against an AP champ your Cloth Armor will have nearly no effect and you are going to have a very hard time if you choose to stay in lane. If you decide to swap lanes too it will set you back because you will miss Experience and Creeps.