Claiming something belongs to you is fairly straightforward in French, once you remember that everything is either masculine or feminine.  C'est mon stylo – It's my pen.  C'est ma nouvelle voiture – it's my new car.  Ce n'est pas ma faute – it's not my fault..etc  If it's more than one thing you're talking about it's simply mes: Ce sont mes enfants – they're my kids.  Mes parents habitent à Eastbourne – my parents live in Eastbourne etc. There's a slight hitch if you're talking about something feminine beginning with a vowel.  You can't say ma_actrice préférée as it sounds too mushy, so you borrow the mon shape and use that: mon actrice préférée – my favourite actress.

If you are talking to a child or a close  friend about something that's his or hers, you use ton,ta or tes in exactly the same way.  Où est ton papa? (Where is your dad) Que dit ta mère? (What does your mother say) Met tes chaussures! (put your shoes on) Où est ton épouse (Where's your wife).

In all these examples, the mon, ma or mes or the ton, ta or tes agree with the thing or person that is owned by me or you.  Mon père, ma mère, mes amis etc.

The same is true of son, sa and ses (his or her or one's ....)  Son père could mean his father or her father, or even one's father if you are generalising..  It should be obvious from what went before whose father is being talked about. Sa mère could easily mean his mother. It depends on who you've just been talking about. This is quite difficult, as in English it's the person owning who determines if it's his or her (Here is a picture of little Thomas with his mother – voici une photo du petit Thomas avec sa mère)

Most of the time it is obvious whose is the person or thing referred to.  But there can be ambiguities. One example is the classic wedding photo, with bride and groom and the two sets of parents.  How do you say Well, her mother looks a bit plump? Sa mère could be his or hers.  This is where you have to use another way of saying her to reinforce just whose poor mum you are talking about. Sa mère à elle a l'air un peu dodue!(As opposed to sa mère à lui)

Which brings us on to another way of doing ownership. If you find a scarf lying around and you want to find out whose it is, the easiest way is to shout: à qui est ce foulard? (whose is this scarf?) The answer might be C'est à Irène (It's Irène's). Or the owner might come forward and say C'est à moi! (It's mine).  He'd be using the same Moi as in chez moi (at my house), venez avec moi (Come with me) etc.  The set is made up of Moi (me) Toi (you) Lui (him) Elle (her) Nous (us) Vous (you lot, or you,sir) Eux (them) and Elles (them if they are all feminine)
These are sometimes called strong pronouns, as they stand up in places where they are exposed, instead of being cosily tucked in in front of the verb as pronouns usually are. You'd say Donnez-moi un gâteau (give me a cake) but Ne me donnez pas de mal (Don't give me any trouble)

C'est à qui? Also means Whose turn is it? either in a queue or in a game (C'est à toi de jouer? - It's your go)

C'est à moi is simply for claiming ownership of something in front of you or being talked about at that moment.  When you say mine in English to mean my one there is something a but more complicated in French for that!  That car isn't mine. Mine is green.  Cette voiture n'est pas à moi. La mienne est verte!  Ce foulard n'est pas à moi, le mien et en soie. That scarf isn't mine. Mine is made of silk.  Yes, because you know what you are talking about, Mine (le mien, la mienne, les miens, les miennes) has to agree with it or them.  You might describe it as a minefield!  If you feel up to it, the set is made up of: le mien (mine), le tien(yours), le sien(his, hers, one's), all of which agree in the same way as le/la/les/les mien/ne/s. To these add le/la nôtre, les nôtres (ours) le/la vôtre, les vôtres (yours) and le/la leur, les leurs (theirs)