When use this letter? I saw this letter used in Past tense, idi and imiş, example: Bu bir hastaydi- bu bir hasta idi. Why not hastadi, it´s not vowel to vowel. Or, anladım, why not here anlaydım? Thanks.

It depends on the suffix, when the last letter is a vowel. You just need to get familiar with the suffixes. 

You´ll see it written like this often -(y)... means for that suffix, add a ´y´, after a vowel.

This link shows several different suffixes http://www.turkishlanguage.co.uk/suffix.htm

-den is interesting. If you add it after a vowel and it´s the first suffix, you don´t add ´y´, if you add it after a suffix that ends in a vowel, you do add a ´y´. Try doing that correctly during a conversation, not easy...

You don´t understand me, but ok. Thanks  

Anybody with explanation?

If you or anyone can explain why my explanation does not answer the question, or is not correct, please do so. 

I asked why we write hastaydi and why not write anlaydım. Why in first we add y and in second not.

It´s just the way Turkish grammar is. You are mentioning 2 different tenses. These different tenses have 2 different suffixes. Each suffix has different rules. Although, I agree with you, the 2 you are mentioning are similar.

hastaydi - to be past tense

I was this, I was that, you were this, you were that...

you must add y after vowel:

http://www.turkishlanguage.co.uk/tobepos.htm

anladım - simple past tense of a verb

I ran, I spoke, I knew, you ran, you spoke...

 you don´t add y after vowel:

http://www.turkishlanguage.co.uk/pasttense.htm

It´s a good question you are asking. ... Here´s another one: why do we say bildin mi? and not bildi misin? 

Or in English, why is ´since´ not written as ´sints´?

This is good answer! Thank you, and you said me this is grammar rule why we write anladım and not anlaydım? 

Welcome!

If -di,-miş,-tir,-se suffixes are used after a noun or as a second tense they are called suffix verbs (ek fiil).And you use y buffer letter if it is necessary. 

hasta is a noun. It ends with a vowel so you need to put y if you want to say hastaydı or hastaymış or hastaysa because here -dı, -mış, -sa are suffix verbs (because they come after a noun)

but

anla is a verb so you dont need to put y when you want to say anladım, anlamış, anlasa(because here -dı, -mış, -sa comes after a verb as the first tense)

and when you use two tenses such as "gitmişti"

miş is first tense, ti(-di) is second tense so -ti is a suffix verb but here it doesnt need any buffer letter because it doesnt come after a vowel.

but

when you look at this example "gelseydi", we still see y because -se is first tense and -di is second tense. So -di is a suffix verb and it needs a buffer letter because first tense -se ends with "e" vowel.

Result: If a suffix verb comes after a noun which ends with a vowel you need buffer letter, if a suffix verb comes after first tense and if that tense´s suffix end with a vowel you also need a buffer letter.

I hope you understand me

Yes, I understood, thank you so much! 

Maybe that is because I am a great teacher

Yes! 

You know how to spoil me

There is some more explanation on when to use buffer letter -y here: 

http://www.turkishclass.com/forumTitle_62062

About the use of the buffer ´y´:

http://www.turkishclass.com/forumTitle_61671

1. past to be + adjective: yaşlıydı, he was old

2. past to be + noun: siyah bir arabaydı, it was a black car

3. And furthermore all past* suffixed verbs ending with a consonant. 

- past conditional: olsaydı, if it was

- past necessitive: aramalıydım, I had to call

- past subjunctive: tıklayaydım

*Past being either -di or -miş

Follwing that rule: başla+mış+ım is without y because the verb stem is not suffixed with -sa or -malı or -yayım or such.

And it is başla+sa+y+mış+ım because başla is suffixed with the conditional suffix -sa.

Why we don´t add to anladım? Because anla isn´t suffixed or?

What heppend with (possessive) pronouns? Benim/Ben, how to say It was mine/me?

Anla is a verb. If it was a noun you must add the buffer letter.

anlasaydı(if he had understood)now I added a buffer letter because there are two moods(first one -sa, second one -dı) and -sa ends with "a".

It was mine: Benimdi.

Thank you.