The Executive and Judicial branches are pretty simple but the Legislative branch is a little confusing. (Well, the Presidential election is a little confusing, too. I'm not even going to try to explain the Electoral College.)
The federal government and all the state governments have bicameral (two branch) legislatures, except Nebraska which is unicameral. The two branches are the Senate, where seats are allocated by governmental entity. U.S. Senate seats are allocated 2 per state and have a six year term. Terms overlap so that in any election (every 2 years) 1/3 is being voted upon. The House of Representatives is allocated by population so more populous states have more representatives. All seats of the House are up for re-election every two years. Districts are redrawn after each Census (at 10 year intervals). The rules governing representation of territories (Puerto Rico, Guam, U.S. Virgin Islands, etc.) and Washington D.C. are different. If those details are needed I recommend looking them up.
State governments follow the same general pattern with a Senate and House. The terms and rules for the entities can be a little different from state to state. In Michigan, each county is roughly the same size and is broken into townships. In other states the county sizes are completely different. Louisiana has "parishes" and Alaska has "boroughs" instead of counties. The Executive position at the state level is Governor and in most states they have a 4 year term..
DC mentioned Mayors, which is the most common. Instead of a Mayor, city and town governments could have a City Council that runs the city directly or a City Manager, usually hired by the city council.
