There are two clubs named Manchester, Manchester United (historically more successful) and Manchester City (currently better). Manchester United usually goes by either Man U, Man United, Man Utd or United. Manchester City usually is called either City or Man City.
There are many football divisions in England, all in essence stacked on top of one another. The top division is the Premier League. Below that is the Championship. Then comes League One, followed by League Two, etc.
If you finish in the bottom 3 of whichever division you're competing in, you get demoted, or relegated, down to the division directly below you for the following season. Teams that finish in the top 3 in their division get promoted to the division above them, replacing the teams that were relegated.
Not always top and bottom 3. 3 shift between Premier and Championship, and 3 between Championship and League One, but 4 teams move between Leagues One and Two, and only 2 are relegated out of League Two.
If you finish in the bottom 3 of whichever division you're competing in, you get demoted, or relegated, down to the division directly below you for the following season. Teams that finish in the top 3 in their division get promoted to the division above them, replacing the teams that were relegated.