If you mean your own artwork, then you'd take a picture of it, upload it to your computer (make sure you know where you're saving it), then select Insert > Picture in Word, browse in the location where you saved the photo, click the picture, and insert. If you mean an image from the internet or something, you can just save it to your computer and basically follow the rest of the steps above.
To download the clip art and animated GIFs on this site, just move your mouse over an image or animation and right-click. Then select "Save Picture As..." (or "Save Image As" if you're using a Mac). Feel free to open the pictures in your favorite graphics software, such as Photoshop, Fireworks, Gimp, etc... and edit if you wish. Keep in mind that linking to these images is not permitted.
I'm seriously angry that Clipart is gone. Why? Because many of those cheesy images were available as Vector Graphics. So, I could quickly insert a crappy clipart into my powerpoints, ungroup the elements of it and delete all but what I wanted. I used it and it worked awesome for me multiple times. I wrote clipart off for years and only started doing this in the past 2 years...I wish I had known earlier.
This is TERRIBLE! Besides being able to edit vector images like Will mentioned (which is important for those of us who don't have graphic artists available to create custom images or the budget or time to buy that service ad hoc), you could also search for style "families" (different graphics that were all in the same style). There was a huge range of files and styles in the collection and it was much faster to find what you need than to do a Bing search. Plus, you knew that everything there was free and free to use, without having to worry about CC license issues.