Rank these houses in order of importance to the story

[Nictarion]

Stark, Lannister, Targaryen. Their members have the most POV chapters and were POVs since the first book.

This.

[Gurkhal]

For me its these ones. I feel its kind of obvious that the Lannisters, Starks and Targaryens are the main Houses in the series and each of them have their side-kick House, Lannister-Tyrell, Stark-Tully and Targaryen-Martell. Possibly the Baratheons will team up with the Greyjoys (Asha and Theon).

Lannister - they are the engine which kind of put the story into progress

Targaryens

Starks

Baratheons

Greyjoys

Tyrell

Martell

Tully

Arryn

All the rest of them.

[Lion of Judah]

Starks then Targaryens and then everyone else.

[KOM]

Starks then Targaryens and then everyone else.

Basically this.

Starks=Ice. Targs=Fire.

--

Tullys

Lannisters

The Baelish wacko

Blackfyres

Greyjoys

--

Pounce

Casso

--

Everyone else

[Draxus]

Stark and Targaryen are the two most developed Houses in the story. And they have both had their downfalls and will both have their revivals. And they both need each other. Stark and Targaryen. One of the many Ice/Fire symbolisms is Ice represents Stark, Fire is Targaryen and Jon is the combo of both. And when Jon goes bad, Dany and Bran will save him. Stark and Targaryen. Was it really ever anything else?

[Jon Icefyre]

Well I don't know how I would rank all of them but my top 5 are definitely:

1. Stark
   
2. Targaryen/Blackfyre(expect House Blackfyre to now become a huge issue with Faegon landing in Westeros)
   
3. Lannister
   
4. Baratheon(as long as Stannis still breathes)
   
5. Martell(they've become huge late-game players)
   

[PrimeRhaegar]

Starks and Targaryen without doubt its called a song of ice and fire for a reason. Heck its its there two sigils thats is playing an important role as well.

after them its lannister, baratheon, tully tyrell, greyjoy, martels

[Carl Snow]

• Stark (ice)
  
• Targaryen (fire)
  
• Lannister (game of thrones)
  
• Baratheon (the engine)
  
• Greyjoy
  
• Tyrell
  
• Bolton
  
• Frey
  
• Martell
  
• Tully
  
• Baelish
  
• Clagane
  
• Arryn
  
• Mormont
  
• Tarly
  
• Umber
  
• Karstark
  
• Dayne
  
• Royce
  
• Harlaw
  

[Corvo Attano]

Obviously, house Tollett is the most important.

[Morienthar]

Targ

Starks

Lannisters

Baratheons

Tyrells

Martells

Greyjoys

Arryns

[Northbound]

The series began, in the North. The first POV was of a Stark. The series will most likely end in the North. STARKS.

[~DarkHorse~]

They are all important in their own way. However, I'd say the ones we had from the very start are still most important for the end result of the story- Starks, Lannisters, Targaryen and Baratheon.

[Fat Mac]

1. Stark

Initial protagonists of the story. All Starks that have had POV have had a significant amount of chapters.

2. Lannister

Initial antagonists. All of Tywin's children have had considerable POV arcs, and other members such as Tywin and Joffrey have had large impacts on the story.

3. Targaryen

The first POVs that didn't take place in Westeros were that of Daenerys. She has her own independent storyline that could possibly represent the fire aspect of "Ice and Fire"

4. Baratheon

Five different Baratheon's claimed kingship in Westeros, with two of those still around. Stannis remains one of the largest and most important non-POV characters in the story.

5. Tyrell

Only really kick in around book 3, but since have had a major impact. Much of Book 4 revolved around the Tyrell-Lannister alliance and it slowly but surely falling apart. Sadly, no POV yet but that doesn't mean the Tyrells aren't important.

6. Greyjoy

Two Greyjoys claim kingship. Four separate POVs for Greyjoys, one with a significant amount of chapters (Theon). Greyjoys all over the aSoIaF world, in their own story arcs.

7. Martell

Like the Tyrells only kick in about book 3. Oberyn was huge for Tyrion's story and Doran/Arianne are huge for the events to come.

8. Tully

This is if you consider Cat a Tully. Tully's played a large role early on with the War of Five Kings, and Riverrun was commonly seen. Lysa played a key role in Sansa's storyline, as well as getting the whole Lannister vs. Stark ball rolling.

9. Bolton

Key to the Stark storyline, as well as the downfall of the Starks. Key for Theon's storyline, and now Stannis'. Two members make a large impression.

10. Frey

Similar to the Boltons. Needed to turn the tides for Book 3, and had a relatively prominent part in the second half of FfC Jaime plotline.

11. Arryn

It was Jon Arryn's death that started it all, must not forget that. Also, now that Sansa is in the Vale Sweetrobin and HtH will probably gain importance. However overall they are the least important of the Great Houses, mainly due to lack of members.

12. Clegane

The Hound and the Mountain were largely important for the first three books, both interacting with Arya's story. One or both may become important again.

13. Mormont

Jorah and Jeor very important for Dany and Jon's story, respectively. Mormont's played important role as Robb's bannermen.

14. Royce

The series started featuring Waymar Royce. More importantly the Royce play a key role in Sansa's FfC story, and most likely that role will get bigger in WoW.

15. Karstark

One of the many families that led to the downfall of Robb Stark. Now play an interesting role in the War in the North.

16. Umber

Mainly play a role for Greatjon, and now for their split forces in the War in the North.

17. Tarly

Wish I could use Sam's storyline as means to rank them higher. But ultimately, Sam's story has little to do with him being a Tarly. Randyll will now become very important on the KL scene.

18. Dayne

While cool, the Dayne's are overall unimportant except for backstory. No living Dayne besides Darkstar has relevance, and Darkstar's importance seems to be limited to just Dorne.

19. Harlaw

Meh, the Reader is cool but ultimately the Harlaws overall are unimportant. May change as Aeron is trying to raise sails against Euron.

[Juli Greyjoy]

I think you can't rate the houses based on their importance in the story. Every house is important in it's own way and without it the story wouldn't be the same. Sure, Stark, Lannister, Baratheon, Targaryen and Greyjoy are the main protagonists, but just imagine there wouldn't be a house Clegane...

[Maegor the Cuddly]

1. Tyrell (They are the Kingmakers in this situation after all...)

2. Starks (Preparing to make a comeback)

3. Baratheon (If Stannis is still alive and kicking. If he isn't, then remove them from the list entirely.)

4. Martell (They are one of the last 2 unaffiliated houses with an army that can still be used.)

5. Arryn (Same as above.)

6. Targaryen (Potentially making a comeback.)

7. Baelish (If LF decides to make good on his status as Lord Paramount of the Riverlands... If not, he can be removed from the list.)

8. Bolton

9. Manderly

10. Karstark

11. Blackfyre (If Aegon is one)

12. Velaryon (In the way of Aurane Waters)

13. Lannister (They are a spent force. All they have left is money, money that may end up being used to pay off the IT's debt.)

14. Tully (See Lannisters minus money and minus castles)

15. Frey (Despite still holding the Twins, they are an international pariah.)

16. Greyjoys (Unless Vicatarion succeeds, these guys are worse than hopeless.)

Edit: I interpret the OP to mean "Important going forward", there's an entirely different list for important in the past.

[BranRaven]

No mention for Manderly, Hightower, Reed or Florent? All pretty important houses it seems.

Targaryen

Stark

Lannister

Baratheon

Martell

Greyjoy

Tyrell

Tully

Bolton

Frey

Arryn

Dayne

Clegane

Mormont

Tarly

Umber

Royce

Karstark

Tarly