And this is how I am going to remember the word птица. http://www.creative.su/sites/creative.su/data/XmUserFiles/userfiles/user9083/bezymjannyjj78785d.jpg
Мой is a masculine form, моя is feminine. (I can see you're learning Portuguese, so мой = meu, моя = minha.)
There's also a third form, моё, of neuter gender.
Now what you do when the bird is actually male I couldn't tell you.
If this is absolutely crucial to tell the bird is male, you could add an apposition, «моя́ пти́ца-саме́ц», or something like this.