Removing dimmer switches, 4 wires
Sounds to me like the white wire is the ungrounded (hot) conductor, and should be reidentified as such using a bit of black tape/paint/marker. Power comes to the first switch via the white, and then to the second with a white jumper.

The red wire is the switched "hot", which is used to control one light. The black wire is the other switched "hot", and is used to control the other light. The bare wire is ground.

To install two new switches:


Turn the power off, and verify that it's off.
Mark the white wire with a bit of black tape/paint/marker.
Connect the white wire with two short bits of black wire (pigtails), using a twist-on wire connector. 
Connect one black pigtail to one switch terminal, and the other to one of the switch terminals of the other switch.
Connect the red wire to the other terminal of one switch.
Connect the black wire to the other terminal of the other switch.
Using bare copper or green pigtails, connect a ground wire to the green screw on each switch, and the box if it's metal.


When done, it should look something like this.