Quick reminder
To make: Assets → library → files icon → New Library Item
OR: select item and drag it to the library window
To use: Drag from library window on to your page
Library items icon: 
found in the Assets menu
Details
Libraries contain page elements such as images, text and other objects that you want to reuse and/or update frequently throughout your web site. These elements are called library items. They can be words, sentences, whole paragraphs or single characters.
When you make a library item DW makes a file (.lbi) and keeps it in a folder (called library) in the root folder ready for reuse. The library folder is made by DW when you make the first library item. If you alter the library item all the instances of it used on your site will change.
See DW Help: Working with Library Items
To make a library item:
Select the element you want to be the library item. In Files panel choose Assets, then the library icon (looks like an open book). This opens the library window. Then go to the files icon (on files panel, top blue bar, it is on the far right and it looks like a bulleted list) there choose New Library item. The item is made and called untitled. Change its name to something useful to you.
To use a library item:
You can drag it from the library window on to your page. (You cannot drag it from the library folder.)
Another way to make a library item:
If you already have a library folder you can select a new item from a page and drag it to the library window. DW makes it into a file which you then should rename.
To Edit a library item:
Open it from the files panel and edit it. Note that the change will appear in all instances where you have used that item.
To detach a library item from a page:
You can detach a library item from its link so that it does not change if the library item is edited. It will remain on the page. To do this: select it and click on detach from original on Properties Inspector bar at bottom of screen. You can then edit it if you wish.