Like me for example, I have some traits from some houses but lack other traits. I'm definitely not a Gryffindor because I lack bravery, not a Hufflepuff because I'm lazy, not a Ravenclaw because I'm not creative enough, and not a Slytherin because I'm not pure-blood. What would I or someone who doesn't fit in all respects of each house do? Can you be accepted into a certain house because you have some of the skills of some houses?

I looked at several websites, Wikipedia and Quora can someone explain in layman's terms what would happen to someone who doesn't fit?

You can always fit in a Hogwart House.

The houses don't put you where you have your best traits, but what traits you value more. That's why Hermione and Neville were put in Griffindor, for example: Hermione would be more a Ravenclaw and Neville a Hufflepuff, but both of them valued bravery more than anything else. Not being wise or intelligent is not a "NO" for entering in Ravenclaw: it's being apathic that would stop the Sorting Hat to put you in the house.

Plus, the major traits are not the only traits that are "allowed" in the houses: Ravenclaw values wisdom, intelligence, creativity, and many more, and not having one of them is absolutely not a reason to not be in the house.

And a little addition: you don't have to be a pure-blood to be in Slytherin. Major examples? Snape, Tom Riddle, Harry Potter (he was supposed to be sent there). And after the battle of Hogwarts, I think that there maybe muggle-born that were sent there (but not sure). To be in Slytherin, you need ambition, ruse, intelligence and others.

Don't worry you'll end up somewhere, and you will probably have a say in the matter.

1. The Sorting Hat takes your choice into account as we see with Harry and his "not Slytherin" mentality.


2. 
   

   Hatstalls still go to a house.

   
   
   

   
   

   An archaic Hogwarts term for any new student whose Sorting takes longer than five minutes. This is an exceptionally long time for the Sorting Hat to deliberate, and occurs rarely, perhaps once every fifty years.

   
   
   

   Of Harry Potter’s contemporaries, Hermione Granger and Neville Longbottom came closest to being Hatstalls. The Sorting Hat spent nearly four minutes trying to decide whether it should place Hermione in Ravenclaw or Gryffindor. In Neville’s case, the Hat was determined to place him in Gryffindor: Neville, intimidated by that house’s reputation for bravery, requested a placing in Hufflepuff. Their silent wrangling resulted in triumph for the Hat.

   
   
   

   Hatstalls by J.K. Rowling on Pottermore

   
   

   
   


3. 
   

   It is not the qualities you possess, but the ones you value (aka the Hermione Rule).

   
   
   

   
   

   “I’m not as good as you,” said Harry, very embarrassed, as she let go of him.

   
   
   

   “Me!” said Hermione. “Books! And cleverness! There are more important things – friendship and bravery and – oh Harry – be careful!”

   
   
   

   Harry Potter and the Philospher's Stone

   
   

   
   

In that case, they’ll likely be in Hufflepuff.

It’s unlikely that too many students don’t fit into any Hogwarts house, since most students are likely to have or value certain traits enough to make them fit into one house better than the others, or to have a preference on which house they’ll be in. However, if they truly don’t fit into any house, or fit equally into all the houses and don’t have any preference, they’d likely then go into Hufflepuff. The houses are based on what the founders wanted in the students they teach, and Helga Hufflepuff was willing to teach all types of students.

“Said Hufflepuff, ‘I’ll teach the lot,

And treat them just the same.”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)}}

The Sorting Hat was intended to Sort the students into their houses in the same way the Founders did, and Helga Hufflepuff was willing to take the students none of the other founders chose. Because of that, it’s likely that the students who don’t fit anywhere would still go to Hufflepuff.

“Good Hufflepuff, she took the rest,

And taught them all she knew,

Thus the houses and their founders

Retained friendships firm and true.”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)}}

Therefore, it’s very likely that if any student doesn’t fit into any of the four houses, or they fit equally well into more than one house, they’d likely be just put into Hufflepuff as Helga Hufflepuff accepted any student.

Congratulations! You are just like Harry Potter himself. When Harry was about to get sorted he had the following thoughts:

Harry smiled weakly. Yes, trying on the hat was a lot better than having to do a spell, but he did wish they could have tried it on without everyone watching. The hat seemed to be asking rather a lot; Harry didn't feel brave or quick-witted or any of it at the moment. If only the hat had mentioned a house for people who felt a bit queasy, that would have been the one for him.

A horrible thought struck Harry, as horrible thoughts always do when you're very nervous. What if he wasn't chosen at all? What if he just sat there with the hat over his eyes for ages, until Professor McGonagall jerked it off his head and said there had obviously been a mistake and he'd better get back on the train?

So here we have it. The great Harry Potter, The Boy Who Lived, The Chosen One, also thought that he didn't have any of the qualities that the houses were looking for. He seemed to think that the hat would be unable to place him anywhere, and he would be sent home.

But in the end the hat managed to dig deep inside his mind and find some Slytheriny qualities. So it's likely the hat could do the same for you. As far as we know there is no student that the hat was ever unable to place in a house. Moreover, even if there were absolutely no qualities that you had, and the Sorting Hat had no way to decide where to place you, it is still possible for you to get sorted. Recall what Harry told his son Albus in the Epilogue of Deathly Hallows:

But if it matters to you, you'll be able to choose Gryffindor over Slytherin. The Sorting Hat takes your choice into account."

"Really?"

"It did for me," said Harry.

So if you're sitting there under the Sorting Hat and it's drawing a blank, if you hear that little voice saying "I have nowhere to put you because you have no qualities", then your best bet is probably just to beg the hat to put you in a particular house, and pray that Harry was correct that it really does take your choice into account.