The US is a republic system -- that is voters elect representatives to represent their views in Congress.
The fact that there are two houses with different methods of representation does not change the fact that we are a democratic republic. The different systems in the US is to balance the power of states with large populations and less populous states. Because populations of different states grow or shrink at different rates, Article 1, section 2 of the Constitution requires that the House seats be reapportioned among the population every 10 years. Over the years, however, there were inequities within states whose electoral districts might be divided disproportionally. The Supreme Court in 1962 and 1964 issued two major cases that challenged within-state disproportionate representation and mandated the "one-man, one-vote" rule. See Baker v. Carr, 369 U.S. 186 (1962), and Reynolds v. Simms, 377 U.S. 533 (1964). That's not to say that representation is fair, just yet -- there is still manipulation of election districts, i.e. gerrymandering, by state legislators to keep their party in power.
American's vote for individuals to represent them in the House and Senate of Congress. That contrasts to countries where voters vote for parties, rather than individuals, although they may have a personal connection with a party leader on a numbered list of possible members of parliament. In these countries, known as having proportional representation, a party getting 26 percent of the vote will have the top 26 percent of its list seated in parliament. In other words, if there are 100 seats, that party would seat the party leader and the next 25 persons on his list.
The biggest difference between the two systems is timing. Politics anywhere involves the building of coalitions that can control a majority of elected members of Congress or parliament. In republics, like the US, the system favors that coalitions be formed before the election. Party conventions draw up party platforms in which they state where they fall on various issues. The party's elected members, however, are not bound by the platform to the extent it contradicts or offends voters in their districts or states.
Parliamentary systems, however, have parties that are already bound to certain issues and in some countries an individual party never wins a majority of seats. But the one having the plurality tries to work out deals with other parties and their special interests. Also they have strict agreements among party members in the parliament that require them to vote with the party leadership. The coalition agreements, too, will require that all members vote with the coalition leaders. When the governing coalition disagrees and fails to win a vote, it is said to "fall," and new elections are called.