The expression to shoot a bird or the bird at some one is referencing a rude gesture usually flipping the middle finger at some one. It originated from archers who would pluck their bows with their middle fingers. It's not a very nice thing to do.
The middle finger, long finger, or tall finger is the third digit of the human hand, located between the index finger and the ring finger. It is usually the longest finger. It is also called the third finger, digitus medius, digitus tertius, or digitus III in anatomy.
In Western countries, extending the middle finger (either on its own, or along with the index finger in the United Kingdom) is an offensive and obscene gesture, widely recognized as a form of insult (colloquially known as "flipping the bird", "flipping someone off", or simply "giving the finger"). Hospitality Recreation