I get really confused about the difference, it has something to do with the division of sovereignty & power I think and I can't really find a clear answer on the internet.
In a federal system, a single organisation is divided into parts. Each part is given some degree of autonomy. The powers of the local governments are defined and limited by the constitution of the federation: Power flows down from the strong central government to the local governments. Australia, USA, and Russia are examples of federal governments.
By contrast an intergovernmental organisation is a union of several governments. The powers of the central organisation are defined and limited by the member organisations: Power flows up from the local governments to the central government. Examples of intergovernmental organisations are the Commonwealth, ASEAN and the EU.
Of course there is overlap and a fuzzy edge. During the early years of the United States, the degree to which the USA was a federal nation, or a Union of divers states was the central matter of debate. The move to qualified majority voting in the EU gives it more of a federal character, and there are those in Europe who would like the movement towards federalism to be speeded up (and of course plenty who oppose it).
In typical federal systems, the central government has funding sources that are independent of the local governments. In the USA there are federal taxes. Whereas the EU is dependent on contributions from member states.
A test of how much an organisation is a federation is how difficult it would be for a local government to leave the union. There is no mechanism for a US state to leave the USA. The mechanism for leaving the EU is apparently simple, but complex in practice. For a country to leave the Commonwealth, they just have to say so.
There is one main difference between the 2 systems.
In Federalism, the Federal state is sovereign. And it delegates some of his power to subdivisions that can be states/landers/region. The subdivisions are not subordinate of the federal state. They are at a different level, with different powers but equal in rank. Typically, the federal state is in control of the army, the foreign policy, nuclear weapons, ect ... The subdivisions typically handle some internal matters. Also citizens living in subdivisions are citizens of the federal state.
In intergovernmentalism, the member states are sovereign and primary actors. They delegate some of their power and resources to the central government. In this particular case, you can have common foreign policies while countries are still able to have foreign policies of their own. Citizens are typically citizens of their own countries. Member states are usually ranked higher than the central government.
One example of intergovernmentalism is Confederation. In case of the EU, it is debatable whether this still intergovernmentalism or a particular case of federation. There is elements of boths. European law is somehow ranked higher, but it has no army, no real common foreign policies. Some countries are not pure federation or pure confederation. The Kingdom of Belgium is officialy a federation, but it has some aspects of a confederation.
To answer your question, I would say an Intergovernmentalism is an umbrella term for Confederate and Federal states, as well as closely working regional standardizations (the most neutral way to characterize the EU is that it's an entity that regulates trade deals among the member states.). I would argue that EU is more of a very loose Confederation but more on that in a moment.
Federal States happen in one of two ways: A.) A Government system that is created when a group of sovereign states cede power to a created Federal State (The United States) or B.) the reverse, a Unitary State cedes powers to regional governments but retains some sort of power among them. At either way, in a Federation, laws that are passed by the federal government are automatically binding for all member states, because the federal government is the member state's deciding representative with regards to that power as a state.
Conversely, in a Confederation, member states typically retain full sovereignty. Typically if the Confederation passes a law, each member state's legislature must recognize the law. In essence, Confederation level laws tend to be in the form of treaties between the member states, and not laws that pertain to individuals, which are still in the member state's preview. By this respect, member states of a Confederation retain the right to succeed from the confederation, if they so choose, where as the member state of a Federation does not have this authority (In the United States, the states do not have that right and there was a big War about that contention). Most Confederations are rather transitional and will become Federations as time progresses (Prior to the Federal State of the United States, its member states were organized under the Articles of Confederation... which were very poorly written and left the Confederation Government with little power to solve the various crisis).
I maintain that the EU is a Confederation because of the nature of their member states with relation to the EU. The EU does not act as a sovereign externally and the various member states conclude diplomacy outside of the authority of the EU in some occasions. Again, a distinguishment between confederation and federation is that a member state can secede, as we saw with Brexit.
In all cases, to what level a Confederation and Federation state has power can vary, but the idea is that smaller levels of government typically have more control over peoples lives, where as the larger ones tend to have some control over member state relationships. For example, the Confederate States of America's constitution was largely identical to the United States Constitution (the big differences were that the President served a single 6 year term, had a line item veto, gave the states the right to secede... and protected the rights of slavery lets not forget that.) which meant that the National level government had some similar powers to the United States Federal Government. Conversely, the EU largely retains controls over internal matters, mostly related to trade and movement, and leaves most of the rest up the member states.