There are a few ways that Macro Photography can be defined, Micro-Photography, Macro-Photography and Close up photography.
Micro-Photography and Macro-photography usually refer to the same thing. Companies such a Nikon use the term macro and micro on the same lense, whereas Cannon calls them macro lenses. Below is a screen grab from Nikons own site which show them using both terms.
What does the terms macro/micro refer to?
The term Macro means big and the term Micro means small. If the item you are taking a picture of is small and you want it to come out as larger in the picture, then you are taking a macro shot of a micro subject.
Macro photography is generally done with a dedicated lense which you change out on your DSLR camera (Such as the nikon lense shown above). Below is an example of the detail that you can achieve from a macro lense.
The photo below is an example of a close up photograph which could have been taken with a regular lense or a macro lense from a farther distance.
Other than using a macro lense there are a few different ways that you can produce a macro photo, such as a lense reversing kit, Extension Tubes and Macro Bellows.
Lense Reversing Kit, This is a piece of equipment that sits between the lense and the camera which turns the lense around allowing you to zoom in on an subject more.
Extension Tubes, This is another way to move the lense farther away from the camera but instead of having the lense reversed it puts the lense further away allowing you to zoom closer to the subject.
Macro Bellows, Similarly to the extension tubes the lense is put farther away and can be moved according to the distance required.