I've found that a quick way to cut objects from their backgrounds is by using a marquee tool variant, such as the Magnetic or Freehand lasso, or of course the Magic Wand tool. However, these often leave jagged edges around objects, or when using feathered selections a solid anti-aliased edge that work well with a monochrome matte, but ill-suits transparency.
By using the Pen tool to create a vector path surrounding edges of the object you can accurately pare down the jagged edges or too-soft anti-aliasing of a layer. I've sourced the below info from my Super User answer, with the addition of the above use case.
- Select the layer you want to erase content from
- Draw a path with the Pen tool
- Open the Paths pane (available from the Window menu if it's not visible)
- In the top-right corner of the Paths pane is an options menu, click it
- Select Stroke Path... from the menu
- From the dialog's dropdown select the Eraser tool, click OK
If the result isn't quite what you expect, undo the last action, select the Eraser tool from the toolbar (or hit default hotkey e) and change the brush style and size accordingly, repeat steps 4-6.