Pronunciation is extremely important in spell casting.
That is covered during the Wingardium Leviosa class, in the first book.
Professor Flitwick himself tells the students to be extra careful when pronouncing the spell, and he mentions the example of wizard who got badly injured from wrongly pronouncing the spell he wanted to cast.
A while later, Hermione points out how Ron's spell is being ineffective due to his mispronunciation.
“Wingardium Leviosa!”
he shouted, waving his long arms like a windmill.
“You’re saying it wrong,” Harry heard Hermione snap.
“It’s Wing-gar-dium Levi-o-sa, make the ‘gar’ nice and long.”
“You do it, then, if you’re so clever,” Ron snarled.”
(As rightly stated above by Cearon O'Flynn) And even sometime after that we see a practical example of Professor Flitwick's story, when Seamus feather's burst into flames (scene that was also in the movie).
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Essential (with a caveat)
In the first book, Professor Flitwick explicitly says that speaking the right words is essential for the spell to function correctly.
Indeed, major errors, such as changing an entire phoneme, can have disastrous consequences: "And saying the magic words properly is very important, too -- neverforget Wizard Baruffio, who said 's' instead of 'f' and found himselfon the floor with a buffalo on his chest."
— Harry Potter and the Sorcerer's Stone
If someone were rendering incapable of uttering a spell correctly, they would also be incapable of using that spell.
At worst, it would produce some alternate effect, as indicated in the previous quotation.
At best, the spell would simply fail, as seen when Neville tried to perform magic with a broken nose: “ STUBEFY! ”
shouted Neville, wheeling around and waving Hermione’s wandat the oncoming Death Eaters.
“ STUBEFY , STUBEFY! ”
But nothing happened — one of the Death Eaters shot their own StunningSpell at Neville; it missed him by inches.
— Harry Potter and the Order of the Phoenix
The caveat:
talented witches and wizards can perform nonverbal spells.
A sufficiently skilled spellcaster can use any incantation nonverbally.
Some are even specifically nonverbal, such as Levicorpus .
So someone who had had their throat ripped out by a werewolf (for example)
could still learn to use any spell, but with more difficulty.
It is likely, however, that particularly hard spells would be of reduced effectiveness:
She winced slightly and put a hand to her ribs.
The curse Dolohov hadused on her, though less effective than it would have been had he beenable to say the incantation aloud , had nevertheless caused, in MadamPomfrey’s words, “quite enough damage to be going on with.”
— Harry Potter and the Order of the Phoenix
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Yes the pronunciation is very important For example in The Philospohers Stone Ron
is saying Wingardium Leviosa incorrectly and cannot get the spell to work:
“Wingardium Leviosa!”
he shouted, waving his long arms like a windmill.
“You’re saying it wrong,” Harry heard Hermione snap.
“It’s Wing-gar-dium Levi-o-sa, make the ‘gar’ nice and long.”
“You do it, then, if you’re so clever,” Ron snarled.”
scifi.stackexchange.com/users/46719/Cearon O'Flynn
I believe it's a combination of wand movement (discussed during DA sessions and lessons such as Wingardium Leviosa), intention and knowledge of the spell's incantation and finally the pronunciation of the incantation.
I imagine non-verbal spells require a version of the incantation in one's mind, rather than just thinking "levitate that feather!"It would seem that as the incantation can be dropped, it is the least important and pronunciation is not that essential.
However it is easier to say a spell as well as wand movement and intention, so presumably the less talented or experienced you are, the more it matters.
Also, the relationship between wand and wizard would be less well-developed and instructions would need to be as clear as possible.
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In Order of the Phoenix we see a case where the mispronunciation of the incantation caused someone's clothes to be set on fire, so pronunciation clearly matters: "
Oh no," said Cho rather wildly as he approached.
"
Expelliarmious!
I mean, Expellimellius!
I — oh, sorry, Marietta!"
Her curly-haired friend’s sleeve had caught fire; Marietta extinguished it with her own wand and glared at Harry as though it was his fault.
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I agree that pronunciation is important.
But there are other languages, not to mention those who were very capable of silent casting.
To assume that everyone must use the Latin (?) words for these spells is a bit silly as there were other schools of magic (Beauxbatons and Durmstrang officially mentioned) in the world.
scifi.stackexchange.com/users/49224/Michiyo Komaki
As Ryan Perry already pointed out one answer with the story when Harry used Floo Powder to go to Diagon Alley and said "diagonal" we see "critical or else it will be a problem"
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How important is the actual pronunciation of magic words in  Harry Potter ? To that end, what if there was a deaf-mute Wizard, or a wizard who had his throat ripped out by a werewolf? Or if you had a heavy accent or lisp? Clearly there are foreign wizards, do they have their own spells or translations? Is a Chinese wizard expected to say  Evatacavra ? Also, if it is just the intent of the spell, why would anyone ever say a spell out loud? It's unstrategic.
How important is pronunciation in spellcasting?
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scifi.stackexchange.com/users/47390/Ty Herrell
Pronunciation is  extremely  important in spell casting. Professor Flitwick tells the students to be careful when pronouncing the spell. There are examples of catching fire, getting injured or conjuring buffalo as a result of incorrect p[pronunciation of spells. However, a sufficiently skilled spell caster can use any incantation non-verbally. Pronunciation of a spell, regardless of the language in which the spell is written, is significant in spellcasting, as are wand movement and, less so, intonation. The consequences of mispronunciation may have devastating effects. 
Pronunciation is deemed to be an integral part of spellcasting along with wand movement and, perhaps less so, intention. Mispronunciation can have disastrous consequences when casting a spell. The spells can be in a language other than Latin. 