Definition
The high seas are the parts of the ocean that fall outside any country's territory or authority, a vast open sea that every nation, including landlocked ones, can use freely and equally, much like a shared plaza in a theme park.
The ocean has invisible fences, too
Stand on a beach and look out. The water seems to run in one piece all the way past the horizon. You can't see them, but international law has drawn careful lines across the sea.
About 22 kilometers (12 nautical miles) from the coast is a country's 'territorial sea.' It's like that country's front yard. Its laws apply there, but it can't stop other countries' ships from passing through peacefully (this is called innocent passage). Beyond that, out to about 370 kilometers (200 nautical miles), is the 'exclusive economic zone.' In this zone, the coastal country has the right to fish and to mine the resources under the seabed.
Keep going past all of those fences, and you finally reach the high seas, which belong to no one. They cover about 64 percent of the ocean's surface and nearly half of the entire planet's surface. That is enormous.
So most of the ocean isn't any one country's private property. It's a public space and a shared highway for everyone on Earth.
A sea that's open and fair to all
The biggest rule on the high seas is the 'freedom of the high seas.' No powerful country can claim this open water as its own or block other countries' ships from using it.
On the high seas, anyone can sail a ship or fly a plane overhead without asking permission. Every country also has the same freedom to fish in deep water, as long as it follows conservation rules, and to lay undersea communication cables on the seafloor that connect the continents.
Landlocked countries like Mongolia or Switzerland get the same benefits. Even with no coastline, they can register a ship under their own flag and sail the high seas on equal terms with any maritime nation.
The deep seabed below the high seas is treated differently. Its mineral resources are considered the common heritage of humankind, so the International Seabed Authority helps manage them to keep any one country from taking everything. One exception: where a continental shelf extends past 200 nautical miles, the coastal country may have rights to the seabed there.
To be a bit more precise
Just because no one owns the high seas doesn't mean it's a lawless zone. Every ship on the high seas has the nationality of the country where it's registered (its flag state), and in principle it follows that country's laws and authority.
What happens on board, including crimes, is generally governed by the law of the flag state (the 'flag state principle'). For example, if something happens on a ship flying the South Korean flag, even in the middle of the South Pacific, Korean police and courts have the right to investigate and try the case.
Serious international crimes that threaten everyone's safety, like piracy and the slave trade, are the exception. Any country's warship has the right to seize a pirate ship and even put its crew on trial. For other grave suspicions, such as the slave trade, a warship can approach and check the vessel (a 'right of visit'), but punishment is generally left to the ship's flag state.
In the end, the high seas aren't orderly because someone owns them. They stay peaceful because countries have agreed to respect each other's flags and rules. It's a shared commons for the whole world.
Common misconceptions
Since no one owns the high seas, no laws apply there. It's a lawless zone.
The 'flag state principle' applies: a ship follows the laws of the country whose flag it flies. Also, any country's warship can seize and punish pirates, and ships suspected of things like the slave trade can be approached and checked (a right of visit).
Where you meet it
The middle of the Pacific Ocean, where big cargo ships from many countries sail freely without any single country's control or permission.
The open sea where a cargo ship flying the flag of landlocked Switzerland carries goods all around the world.
In one sentence
The high seas belong to no single country. As long as ships follow the laws of their own flag state, every nation in the world can use this shared ocean freely.