"I see a dog."

Translation: Widzę psa.

[Jesuis89]

Can I say widzę pies?

[tadjanow]

No. Direct object requires (usually) the accusative case - 'widzę psa' is an example of this.
In negative sentences and after some verbs you use the genitive case, but you never leave it in nominative.

[George215793]

Why so many names for a dog( pies) in polish?
What becomes of "Mężczyzna"?

[Jellei]

It's just one 'name' in different grammatical cases. For "I see a man", it would be "Widzę mężczyznę".