Hello Vicky, this is David. I'm glad to see your writing about your lovely cat. I'd like to correct it. You can point out my mistakes, too. ^_^

I had always wanted a cat. (The past perfect tense will show your yearning for a long time)  

One day my mother brought me a small kitten from her friend. (also you can say ''a friend of her'') 

Then (I think it better to use ''at that time'' make it clear) it (If you know its gender you should use ''he'') was a small ball of wool fur/hair (''wool'' refers to sheep, but ''fur'' or ''hair'' can be used with animals in general)  with blue eyes, who (you see? You used ''who'' to indicate the dog's gender, so you should use ''he'' rather than ''it'') was placed on the palm.

For a long time I couldn't think of his name. For two days I just called him "cat". But later I with my friend decided (or ''I decided with my friend that...'') he fit the name Stepash. (or ''Stepash fit him well'') 

She also has a cat and his name is Stepan.

The next three months, the kitten grew up and the colour of his eyes changed to green. Now he's a big cat with green eyes. He has red-and-white hair/fur.

I really like the combination of his red hair and green eyes. (very good!)

He's already almost a year old.

Every day I give him porridge and tidy up his toilet. (I do the same thing! But it's hard for me to clean up their poop, lol)

Once a week I comb him, especially his big, beautiful red tail.

In gratitude (or ''with gratitude''), he sleeps with me every night. (Wow, how kind you are! I don't dare to sleep with them!)

He priholdit (?) me in bed and sticks his wet nose on/onto my cheek. (Gosh! How intimate you are!)

He is very spoiled and loves to play. (also you can say ''loves playing'')

Sometimes I scold him for it because he scratches me and the furniture.

In summer he sleeps almost the whole day.

I think it's because he's hot.

Sometimes I walk him to play in the street.

At first he is afraid and does not want to go, but then I cannot leave him at home.

He really likes the street.
He is very gentle and sweet.

I love him very much and I think he is the best.

Way to go, keep it up!

Faithfully yours,

David

Correct this please!! (better with: Please correct this)

[Reasons: http://www.italki.com/answers/question/74853.htm ]

I always wanted a cat. One day my mother brought me a small kitten from her friend. Then it there was a small ball of wool with blue eyes, which was placed on the its palm.

For a long time I couldn't (in formal writing: could not) think of his its name. For two days I just called him it "cat". But later I with my friend and I decided that he fit the to name it "Stepash". She My friend also has a cat and his its name is Stepan.

In the next three months, the kitten grew up and his its eyes changed color to green color. Now it's (in formal writing: it is) a big cat with green eyes. Its wool (what do you mean "wool"? Is it skin??) his is red and white. I really like the combination of his its red hair and green eyes. He's It is already almost a year old (or: It is already almost a year since my mother brought it as a kitten). 

Every day I give (better with: feed) him the cat porridge and tidy up its toilet. Once a week I comb it, especially its big, beautiful red tail. In gratitude, he it sleeps with me every night. He It priholdit (?? -- no idea what this means) me in bed and sticks his wet nose into my cheek. He It is very spoiled and loves to play. Sometimes I scold him for it because it scratches me, and the furniture.

In summer he the cat sleeps almost the whole day. I think it's the reason is that because he's it feels hot. Sometimes I was walking with it him to play on the street. At first he it was afraid and did not want to go, but then I can did not let it to place his stay home. He It now really likes the street.

He The cat is very gentle and sweet. I love him it very much and I think he it is the best.

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N.B.

1) Avoid writing the whole essay into solely one big paragraph.  Break it down into a number of paragraphs for different ideas.

2) English does not have particular male or female gender in animals, unless they are fantastized as a person in stories.  Thus, either "it" (singular) or "they" (plural) is used. 

Please keep up the good work. / Пожа́луйста, продолжа́й хоро́шую рабо́ту. 

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Good day. / Хоро́шего дня.