The bridge is the only component of a guitar that is always in contact with the string.
It transfers the energy of the vibration of the string into the body. The manner in which this transfer of energy occurs influences the tone.
In general, the harder the bridge saddles, the brighter the tone. This means brass saddles tend to give a warmer tone and stainless steel saddles give a brighter tone with more pronounced treble frequencies.
It is also important to note that the bridge is also use to adjust the action and intonation of a guitar. Bridges with individually adjustable saddles give more freedom to get a better set up.