There are two ways of moving people into a new household, depending on if they're moving to an existing household or if they're moving into a brand new household.
Moving to an Existing Household
To move someone to a different existing household, open their person record, and then, where it lists their mailing address, click their address (Not the orange link where it lists their current household.) Then, you'll see a drop down list where you'll see their current household listed. Select their new household from the drop down list and then, click in a blank space to the right of the "New" button. This will save the change for moving them to the household you picked.
Move to a New Household
To move someone to a brand new household, open their person record, and then, where it lists their mailing address, click on their address (Not the orange link where it lists their current household.) Then, you'll see a button that says "New". Click this button and then fill in their new address information. When finished, click to the right of the "New" button in a blank spot, and you'll be asked if you want to change the address for the existing household, or make a new household. Choose to make a new household, and this will save your changes.