I want to "gang" two of these dimmers. I removed the old switches and now I'm staring at a lot of wires! Help me not get electrocuted, please.

Assuming both switches are single pole...
One black wire from connects breaker connects to each switch.
One black wire from light goes to light
http://waterheatertimer.org/How-to-wire-switches.html#gang

If your switches are 3-way... then re-post new question.

Gene

Water Heater Timers Save Money

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http://waterheatertimer.org/How-to-wire-switches.html#gang

Is the neutral wire inside the gang box, or is the neutral wire in the ceiling box?

Gene

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http://waterheatertimer.org/How-to-wire-switches.html#gang

Gene

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

You have 2-gang switch.
Look at following illustration showing gang switches:
http://waterheatertimer.org/How-to-wire-switches.html#gang

Hot wire feeds one screw on first switch, and then jumpers to second switch.
Your jumper wire is connected wrong place.

Find hot wire... there will be 1 hot wire entering gang box.
Disconnect all wires, and separate for testing.
Turn power on and test each wire to bare ground wire... that will show the hot wire.

Sorry it is difficult to post question on fixya, or communicate with people answering free questions.

Gene h

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

The following should help with wiring your dimmer - all the other answers are referring to a car and no help whatsoever!
http://assets1.auroralighting.com/web/sku/66/general/instructions.pdf

Copy following link for switch wiring illustrations
http://waterheatertimer.org/How-to-wire-switches.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

with a three way switch like this, you will have two traveller wires and a common wire. Look in the instructions for the switch or on the back of the device itself to see if there is a wire labeled "common" or "com". This common wire from the old switch hooks up to the common wire on the new switch. I did a bit of searching on the leviton 6602, but couldn't find anything specifically about the three way version (I would venture to guess, however, that the red wire on the old switch is the common). The common on one of the three way switches will always be a constant hot (or power) wire...the common on the other three way switch will be the "switch leg" or the wire that connects directly to the light. The two travellers go to the other two wires. Check the pamplet that came with the lutron device, it should show which wire is common and which are travellers.

Open following links for images of 3-gang and 4-gang single-pole switches.
With the 4 gang, just subtract one of the switches.
http://waterheatertimer.org/images/Single-pole-switches-in-4-g.jpg
http://waterheatertimer.org/images/despard-500.jpg

If you have a 3-way switch, or outlet in the box, then add a comment with more exact information about what wires you are trying to connect, what color wires you have, and what you have tried so far. This will narrow down the possible answers and speed help.

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I can help, but I need a lot more specific info. Please use these naming conventions to describe what you have:
G1, G2 -- the two gangs of 4 switches each G1S1 - the first switch in gang 1 L1 - light, appliance or outlet number 1
Describe how each light is controlled as follows: L1 - G1S1 & G2S1 (one light controlled by a pair of 3-way switches) L2,L3 - G1S2 (assumes a normal switch controlling two lights) etc. for all of the L's presently controlled by all 8 switches
Then describe what you want the 6 new switches to operate.

I draw a little diagram may help you...