When a weapon is fired, lets take the M16 as an example, ignited propellent gases follow the bullet out of the barrel causing a flash or flame like effect. This is because the gases that have 'exploded' to propel the bullet from the weapon are not fully expended by the act of firing and react with the oxygen in the air. Because the faster these gases are expanded the quicker the reaction is over, the design of the flash suppressor is such that it allows the gas to be spread more quickly than if it were just coming from the end of the barrel - hence the vents you see in the pictures.

A flash eliminator or suppressor has these vents that spread the gases out of the top and sides of the rifle as well as from the front of the barrel, allowing the gases to escape in a number of directions, thus reducing the volume and duration of the flash. It depends on the type, size and vent location as to how effective it is in reducing the flash overall.

The flash also needs reducing so an enemy cannot easily pinpoint the firer and also to aid the firer in resighting the weapon for further shots etc, because a large flash is distracting. They are also necessary when using some types of NVDs as an unsuppressed flash can wipe out a night sight picture.

A muzzle brake/break is similar to the suppressor but is designed to minimise recoil by disseminating the gases in a similar way. By pushing the gases out in a number of directions there is less rise in the barrel as there is less direct reaction from the bullet exiting. It can be incorporated into a suppressor. Muzzle breaks by themselves can actually increase overall muzzle flash and are usually sole fitted to sporting guns not military weapons.

Neither reduce the noise of the weapon, but they do change it and you can hear the difference in a rifle fired with it's suppressor on and off.