Macro photography is when you photograph something at a ratio of at least 1:1 magnification. Close-up photography is simply taking a picture "close-up" and filling the frame with that object.
There still seems to be debate as to whether or not you can actually shoot a macro shot without a dedicated macro lense. Me? I believe that you don't have to have a macro lense... this picture, for example, was shot using a basic 18-55 mm lense:
http://farm4.static.flickr.com/3228/...b221d34bda.jpg
Here is another using a dedicated macro lense (it is the center of a marigold flower):
http://farm4.static.flickr.com/3454/...057bbb9f8c.jpg
Definitely a difference, but it all depends on what one person's definition of macro is versus another's definition.
Hope this helps!