The outcome is not always very pretty. Word often brings over extraneous formatting, which adds extra spacing, strange characters, or unwanted code to your content. Alas, there is an easy solution: copy your text to a plain-text editor first, then recopy and paste into Blackboard.
Windows users: Copy text from Word to Notepad (already installed on your computer), then to Blackboard.
Mac users: Copy text from Word to TextWrangler (the best free text editor ever, IMHO), then to Blackboard. If you don’t have TextWrangler, you’ll want to download it via that link. Alternatively, you can copy the text into TextEdit, which comes with OSX, highlight your text, and Click Format > Make Plain Text.
This PDF from Grand Valley State University gives complete directions with screenshots for copying into Notepad and TextEdit.