What is the difference between меня and мой?
I understand that both of them are used to denote possession, but I'm not sure how they differ since both of them refer to 'я'.

Question

Answer

Мой is used to tell that you own something.

If you have a pen you can say - It's my pen - Это МОЯ ручка. (fem)
It's my house - Это МОЙ дом. (masc)
It's my jam. - Это МОЁ варенье (neuter)
These are my kittens. - Это МОИ котята (pl)

Меня answers the question - whose - чей?

It's ... father. (father whose - mine - father is masculine - мой)

Меня is a form of the pronoun I
genitive case or accusative case
Меня answers the question whom (for whom and alike)? (кого) 
Ex: My mum loves ... .  (loves whom? me - меня)

You can practise: 
1. Жди ... завтра. 
2. Это ... мнение. 
3. Хочешь увидеть ... ?
4. ... ответ - нет. 
5. Ты узнаешь ... .

Good luck!)

Answer

Rough informal explanation. Because «меня» is a form of «я», into English it will be translated with “I” (but not always vice versa because of its other forms in Russian), while «мой» will be translated as “my” or “mine”.

Formally it can be explained like this:
«Меня» is a pronoun «я» (“I”) either in genitive (also called as possessive) or in accusative case.
«Мой» is possessive pronoun of “my”.

Some examples:
My phone → Мой телефон
I'm being dressed → Меня одевают
I'm absent → Меня нет

Answer

Don't you ask about мой and у меня? Because they have often the same meaning.
У меня отец умер = Мой отец умер. У меня пуговица оторвалась=Моя пуговица оторвалась. У меня голова болит = Моя голова болит. Using "У МЕНЯ" is often more natural.
You cannot use  у меня after это. Это мой отец.

Answer

One more eample:
Do you love ME? - Ты любишь МЕНЯ?
It's MY home - Это МОЙ дом.

Answer

Мой is masculine and моя is feminine. Both mean my / mine.