Electronic – arduino – Cannot get Sainsmart 8 Channel SSR board to turn on sprinkler’s 24 V solenoid

OK, now at least we know what you're talking about.

They are compatible in the way that you can directly connect the Arduino output to the relay module's input. Arduino's logic is 5 V, and the module needs 2.5 to 20 V input to drive the relay. So that's OK.

The Arduino needs at least 6 V input (7 V recommended), but the circuit works at 5 V, and that 5 V is also available on the power connector, at the bottom of the picture.

Connect this 5V and the ground next to it to the relay module's power connector, and up to 8 of the digital I/O's shown at the top to the relay module's logic inputs and you're in business.

edit
vicatcu explains that the Arduino can also be powered from the USB input, and that's true. But the relay module will draw up to 160 mA, and that may be more than your USB port will supply. The Arduino itself also needs around 50 mA.

edit 2012-07-09, re jippie's comment
The Arduino Uno's LDO voltage regulator can supply 800 mA, which should be sufficient for Arduino + relay module, together about 200 mA. At 6 V in the LDO will dissipate 200 mW, and that's no problem, but at 15 V in that becomes 2 W, and that may be too much for the NCP1117's thermal protection. After all this is an SMT device. So it's advisable to use an as low as possible input voltage, or use a separate 5 V wall wart to power the relay module.

According to the datasheet, the G3MB-202P is a discontinued zero-crossing switched AC SSR.

So, when it is commanded to turn on, it will delay by up to 1/2 of an AC cycle until the next zero crossing. So, at 50/60Hz it can delay by as much as 10msec or 8.33msec

When it is commanded to turn off, it cannot turn off until the current crosses zero. This can be as much as 10msec or 8.33 msec again, but not necessarily in phase with the voltage.

For a resistive load you if you commanded it on for a bit more than 10msec it would be guaranteed to turn on, and commanded it off for a bit more than 10msec it would be guaranteed to turn off. Using say 10.5 msec for each, that's 21msec or about 47Hz. In reality you would get beating between commands and response frequencies with wildly varying amounts of power, so perhaps 1/10 of that frequency or about 5 or 6Hz is more reasonable as a maximum, but even that is a bit high.

If you want to get 'even' and fairly-beat free power (for example for a PWM heater control), normally 0.5Hz is about right (gives you 200 half cycles per period at least).

Nothing you can do by commanding the relay will damage it with a simple load like lights, but you may get undesirable (or perhaps interesting) variations in light if you cycle it too fast. Some will probably appear as smooth pulsations in brightness, which might be quite okay. The life of the lights might be shortened a bit if they're incandescent, but they're only Christmas lights.