The existence of a law/legal system is the province of sovereign states. We do not have a world government so there is no world legal system.
There is such a thing as international law, however, that is based on what the sovereign nations of the world agree is international law (usually in a treaty) and the degree to which they have implemented them in domestic law. For example, the International Criminal Court has no jurisdiction over US nationals because the United States of America has refused to ratify the treaty that created it.
There are also supra-national entities like the EU whose directives are binding on their member states and such states are required, as a condition of their membership, to enforce such directives domestically.
A sovereign state's courts will decide when a person and their activities falls within their jurisdiction based on the circumstances of the particular case. For example, an Australian citizen can be prosecuted in Australia for paying a bribe to a foreign official in a foreign country even while working on behalf of a foreign company even if such activity is locally lawful. Why? Because Australia is a sovereign nation and it says it can.
Sometimes it is impossible for a person to comply with the laws of multiple nations. For example, if the EU requires that certain data about their citizens is to be made confidential but the laws of the USA require a US corporation to disclose this information then it is impossible to comply with both. A person in such a position must decide which laws they will break.
It is partly for that reason that multi-national corporations are usually multi-national corporations i.e. they have a different corporate subsidiary in each jurisdiction (tax is another reason). For example, if all EU citizens do business with Google (Europe) then Google (USA) can rightfully claim that it has no data about European citizens to disclose.