After the winter storm when I turned on my water valve for my sprinkler, water gushed from the top. I notice the black top was cracked and I could take it off with my hand. Inside was filled with water and the rubber plastic ring and white and red plastic piece was floating at the top. Is this something easy to fix? 
Sprinkler: Water leaking from the black box with the screws
sprinkler-system
Related Solutions
My guess would be that there is something wrong with the backflow preventer.
Most likely, the air inlet valve is not sealing properly. When pressurized from the side marked "normal flow" (which is where your tap is), it should open the check valve, and also force the air inlet valve shut.
It's probably a rubber gasket, and might be worn out and need to be replaced, or maybe something is physically holding it open. Try pushing on it to see if you can get it to close all the way. Some silicon grease around the gasket will help it form a good seal and prolong the life of the gasket.
First we need a Manufacturer and possibly if you can find it, a model number. Irrigation control valves like this have two parts, a diaphragm operated water control valve and a vacuum breaker. Typical Rainbird system shown here.
The control valve uses an electrically controlled needle valve that operates the main diaphram valve through servo action (small force controls large force). Cross section to give an idea how this works. When the needle is seated, full water pressure operates on the upper side of the diaphragm, shutting water flow off. When the needle opens, the pinhole in the diaphragm is smaller than the needle opening, pressure dumps and the diaphragm opens.
Where your leak is occurring is in the vacuum breaker (right hand side below). This device operates to prevent water siphoning back into the water line through the control valve if water pressure drops. It is essentially a check valve with one side open to the atmosphere that closes under pressure and if the parts are damaged through wear, foreign object stuck in the seat, corroded in place, etc. becomes as you've found, a large volume leak under pressure.
This is where having all that manufacturing information is important as most of these can be repaired if you can find the repair kits. Otherwise, it's complete replacement time.
Exploded diagram of the above example Rainbird unit less the solenoid control.
Champion Classic Brass 3/4" Automatic Actuator and Anti-Siphon with Union
Manufacturer is Arrowhead-Champion http://www.championirrigation.com
I've used Irrigation Direct for other product, their replacement parts page. There are other businesses listed. If you have a large irrigation supply anywhere near, they probably carry the product. Arrowhead is a well known manufacturer of brass plumbing fittings.
The Champion product has been around in manual systems since the 1930's. To get the Vacuum Breaker loose, you might need gentle heat on the body. Keep it on the breaker side.
Breakdown of the original Manual Valve and Anti-Siphon with Union
And the bits in the Anti-Siphon Valve that are giving you grief
Best Answer
That appears to be the backflow preventer and you may be able to find replacement parts for that specific make/model at a local irrigation supply house.
Otherwise, you'll need to replace it.