[Edited] You use just σου if it is to follow the noun as σου cannot stand alone in a sentence; it needs to follow a noun or δικός. Possible combinations with possessives are:
Ο σκύλος σου - Your dog
Ο δικός σου σκύλος - Your own dog (this is a bit emphatic)
Ο σκύλος είναι δικός σου - The dog is yours
Δικός means 'own', and σου means 'your'. One language's words match those in the other in the second example in my comment above (if you just ignore the article). But 'yours', as it stands alone in the third example in English, cannot be conveyed in Greek in the same way, in one word: you use δικός σου for that too.
Only the letter -ν is sometimes removed from the end of some words when the next word starts with a vowel.
When speaking (quickly), δικός σου and δικό σου sound the same, but you understand what the case is by the article just before. You can't omit the -ς from the masculine in written form, though.
"Αυτός είναι ο σκύλος σου" is the most natural sentence and it means "This is your dog" (not "The dog is yours / your own"). The article ο before σκύλος is absolutely necessary here. Αυτό instead of αυτός could be used ("Αυτό είναι ο σκύλος σου"), but it is odd and would require a particular context (e.g. as an answer to the question "Τι είναι αυτό;", made by the surprised owner of the dog).
They may mean the same, but they are very different sentences gramatically. 'είναι' is a form of the verb 'to be', so you can't use the verb 'to belong'. And 'δικός' indicates that 'σου' is a possessive pronoun, but in your sentence 'you' is a personal pronoun.
You would say ο δικός σου σκύλος. Let's say that δικός is an adjective. Then, ignoring the decorative things around it, like the article, it can only go before the noun. You can place it after the noun only if you separate it with a verb: ο δικός σου σκύλος vs ο σκύλος είναι δικός σου, as in your own dog vs the dog is your own or a beautiful flower vs the flower is beautiful. It's basically the same principle. :)