A grounding bar is most often located in the breaker panel. This is where all your grounding conductors are landed. If the panel is your primary service panel, neutrals and grounds can both be landed there.
The NEC (US) requires that all service equipment be bonded together. This includes your meterbox. To most people bonding and grounding look alike but they serve different purposes. In most cases this bonding consists of a #6 AWG solid copper conductor connecting your service panel to your meterbox and also whatever you happen to be using as a grounding electrode, usually a water pipe and ground rod.
Consult a competent electrician for what is required in your area. I was on a job once where the city inspector expected this bonding conductor to be run with the service conductors inside the same conduit, which is what I would expect. But the power company for the same job required it to be run outside the conduit which is acceptable practice. Both were right, but we had to do it one way for the inspection and another way before power was supplied.
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The panel and breaker have to be of the same manufacturer. This way the breaker will fit into the panel. If the panel has a push in bus bar, the breaker must also be the type to accept the bus bar. If the bus bars in the panel are of the bolt in type then the breaker also has to be a bolt in breaker.
Sounds like it is a 220-240 Volt hot water heater. The black and red are connected to the 220 volts supply and the white is connected to Neutral. At the breaker panel red and black connect to the 2-pole 220 volt breaker and white goes to the neutral bus bar.
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Without more information that appears to be a 240 volt circuit. Red & black would connect to the 240 volt breaker, white connects to the neutral bar, and ground conductor from the ground rods or ground plate connects to the neutral bar. Make sure that the bonding screw is in the neutral bar and it is screwed through to bond the distribution panel enclosure to the neutral bar.
Washing machines in the U.S. operate on 120 volts. That requires a single pole 20 amp breaker and wired with 12/2 w-ground wire. Black to the breaker, white to the neutral bus bar, and copper ground to the ground bus bar.