Pasting Attributes is a very handy and simple tool. Final Cut Pro attributes are any filters and/or effects applied to a clip, ie: chroma key filters, speed changes, audio effects, etc. When working with multiple attributes on any clip it allows you to paste all of them to other clips. This works with both Audio and Video attributes.
Note: There are several ways to access the Paste Attributes dialog, I will only mention the default keyboard shortcut.

You may notice that at the top of the dialog there is a Scale Attribute Times check box. This refers any keyframes that you might have on original clip. If you select this, the keyframes will be adjusted to match the new durations of the clips that you are pasting the attributes to.