The main purpose of the bridge on an acoustic guitar is to transfer the vibration from the strings to the soundboard, which vibrates the air inside of the guitar, thereby amplifying the sound produced by the strings.
On both electric and acoustic guitars, the bridge holds the strings in place on the body. There are many varied bridge designs. There may be some mechanism for raising or lowering the bridge to adjust the distance between the strings and the fretboard (action), and/or fine-tuning the intonation of the instrument. Some are spring-loaded and feature a "whammy bar", a removable arm which allows the player to modulate the pitch moving the bridge up and down. The whammy bar is sometimes also referred to as a "tremolo bar" (see Tremolo for further discussion of this term - the effect of rapidly changing pitch produced by a whammy bar is more correctly called "vibrato"). Some bridges also allow for alternate tunings at the touch of a button.
On almost all modern electric guitars, the bridge is adjustable for each string so that intonation stays correct up and down the neck. If the open string is in tune but sharp or flat when frets are pressed, the bridge can be adjusted with a screwdriver or hex key to remedy the problem. In general, flat notes are corrected by moving the bridge forward and sharp notes by moving it backwards. On an instrument correctly adjusted for intonation, the actual length of each string from the nut to the bridge saddle will be slightly but measurably longer than the scale length of the instrument. This additional length is called compensation, which flattens all notes a bit to compensate for the sharping of all fretted notes caused by stretching the string during fretting.