So it's important to note that the Swiss Federal Government Model borrowed heavily from the United States Government model (a lot of other countries do as well, but the Swiss really liked the Federalism aspects and separation of powers). The slight difference is that the executive branch is picked by the legislative branch, not the people, they Executive is a counsel and not a singular person (though the council president is the defacto head of state and government, they are co-equal with the six other members of the council in terms of authority). The courts also lack judicial review (so they can't declare a law unconstitutional).
The chief difference however is the Direct Democracy practices that are enacted at Federal, Canton (equivalent to a State in the U.S. system), and local levels of the government. This is in contrast to the United States, which does not have a Direct Democracy system at the Federal level only (all 50 states have some form of Direct Democracy enabled in their constitution.). The United States Founding Fathers had a big fear of Tyranny of the Majority and felt that while the Government would be controlled by the people, it needed checks against them from depriving rights to the minority. The Swiss decided to use the people as a check on government.
Under the Swiss Democracy, any matter of law or policy is eligible for a Referendum provided that the people can prove it is at issue by receiving 50,000 signatures from citizens within the Federation in a petition. They may also put constitutional amendments up for referendum with a showing of 100,000 signatures. The legislature may "counter-propose" the new law/amendment in question with their own legislation proposal. If this happens both are considered in the next referendum vote.
In a Referendum election, the original proposal is put up for a yes or no vote. If there was a counter-proposal, that option will also be open to a vote and a second question will be asked which basically amounts to "if your choice does not win, would you still be in favor of one of the two remaining options?" to ensure that if you are not going to get your desired way, you can still contribute to a decision on the other offers (If a choice is between A, B, and No and you pick No, if No loses, you would still prefer option B to option A.).
In order to pass, the vote must reach a threshold of 51% for regular laws and 66% for Constitutional Amendments. In addition, the vote must achieve what is called a "Double Majority" such that it must both win the threshold of the Popular vote AND must win the threshold among the Canton level popular vote. This prevents large population centers from fielding a larger popular vote and dominating the Referendum's actions and is similar in practice to the United States Electoral College (which ignores national popularity to state-level popularity if the two are in rare disagreement). Essentially, if 51% of the national population agrees with a bill's passage, but only 48% of the Cantons are in favor, then the bill does not pass. Likewise, if 51% of the Cantons agree, but 48% of the national population is in favor, the measure does not pass.
The net effect of all of this is that the general public act as a stronger check on the legislature in Switzerland than in the United States. One of the advantages to this system used by the Swiss government and not the United States Government is the population sizes. The United States is 40 times larger by population than Switzerland (320 million to 8 million) so the initial petition numbers are more representative of the Swiss opinions than they would be in the United States.
It's best to think of the Swiss Direct Democracy as a "Fourth" governmental check to the other three branches of government in the Swiss Federal system. They have no more authority over any branches than any one branch has over any other branch, but they are an additional part that has to be dealt with in order to enact laws. The net result is that the Legislature is simultaneously answerable to both the nation as a whole and their constituencies and must balance legislation between the two interests to avoid referendums on their laws and losing governing mandate of the people they represent. The Referendum system is merely a tool to allow the people to force an issue that the politicians do not want touch.