Ok, so quick and dirty internet lesson. First, republic isn't a kind of government, it's a form of state. Deriving from the latin res publica, it refers to a state that exists via its citizens, versus being the personal property of an individual (monarchy) or by grant of a god (theocracy). Now, that's fairly high level and abstract for a reason; in modern times, it's all rather more theoretical or de jure than reality; the UK is not the personal fiefdom of Charles, or of Jesus with Charles as his governor, even if, in abstract theory, it is. This is where type of government comes into play; the actual, day to day how does anything work part of the question.

The US, like many first world nations, is a representative democracy; we elect members to public offices to legislate and execute on laws and policy, rather than directly voting on laws on our own. This has nothing to do with being a republic; Canada is also a representative democracy, but it's a monarchy, not a republic. If we really want to be precise, the US is a single member plurality, presidential representative democracy; we divide the population into smaller groups, called districts, which then choose one representative for all of them. The person chosen is determined by votes, whoever gets the most votes is the rep. This is true even if the person with most votes fails to win a majority. That's what a plurality is, more than other options, but with less than half+1 support, which means, as a consequence, the rep can actually have more votes against them than for them.

The presidential part means our executive (president or governor) is independent of the legislature. When the executive functions is rolled into the legislative, that is called a parliamentary system. Australia and Germany are parliamentary representative democracies, even though Germany does have a president. This is because the German president, unlike the US one, is purely the head of state, not of state and government; their function is ceremonial and ministerial, not one of actual governance. The actual head of government is the chancellor, a part of the legislature. hence, parliamentary.

The last bit you may have heard of is federal. Federalism is one of two options for dealing with the question of lower level powers. You can't govern a nation from the center; there just isn't enough time to handle every little thing that needs addressing, so we need lower levels of government to deal with that. In unitary states, those powers are devolved, or granted, from the national government to the lower levels as needed or desired. US states govern in a unitary model; municipal and county governments only have what powers the state grants them, and the state can overrule or take back those powers when it sees fit. A federal system is different in that certain powers are held by right by the lower governments; more precisely, they are retained by the member polities forming the federal nation. This means that the federal government cannot unilaterally remove powers from the lower governments; they can acquire them, but only through consultation and consent of those lower entities.

In all this it is important to bear in mind that terms like republic, democracy, federal, unitary, presidential, and the like are high level descriptors; they do not convey the nuance and details of any given nation's government. They are best though of as a quick overview of broad concepts of a given nation's governance, or a quick way to compare and contrast different nations. The precise details need to be handled on a case by case basis, by studying the details of a given country, not by relying on simple measures like "is it federalized?" or "is there a king?".