Your city council is a group of people who make decisions for your city. In some places they are called a city board, board of aldermen or board of selectmen. In general, city council members are democratically elected and decide which services to provide, how to pay for those services and a lot of other stuff. They work with a mayor or city manager (depending on the municipal system).
Your council is the main law-making body for your city. It can enact ordinances or laws on all sorts of things, from requiring a stoplight to banning DIY guns. It also controls the budget and oversees all the other city boards, like fire, parks, libraries and waste management.
The council typically has a few committees to help with specific issues, such as police or zoning. These committees let groups of council members thoroughly explore the issue before sending a recommendation to the full council. This process helps to reduce the number of meetings and length of those meetings, as well as allowing citizens to participate in committees that address their specific interests.
You might also have a council of representatives, where you elect a number of individuals to serve as your voice on the council. These people might be your district representatives or, in some cases, you might elect a bicameral council. Some cities have political parties, where candidates run as Republicans, Democrats, Libertarians or Greens — and these party affiliations are clearly listed on the ballot when you vote. Other cities don’t have a party, and in these cities, councilmembers are generally considered non-partisan, even if they have views that align with a political party or get endorsements from a party.