What is the difference between "magnification" and "magnifying power"? I've read it a lot of times. But I've not been able to get around magnifying power and magnification of a simple microscope and the difference between them. Can someone explain?
 A: (Linear) magnification is equal to $\dfrac{\text{image size}}{\text{object size}}$.
Problems arise with applying this definition when objects and/or images are a very long way away - usually termed "at infinity".  
It is easier to describe a telescope and then move on to the microscope?
How can you reconcile this definition of magnification if you are told that a telescope focussed on the Moon has a magnification of $10\times$.
Where is this image which is ten times the size of the Moon?

This is where you have to use the idea of a visual angle which is the angle $\alpha$ subtended by the object or image at the eye. (Diagram 1).  
The visual angle gives you an idea of the apparent size of an object.
In Diagram 2 although object $O_1$ is larger than object $O_2$ they both appear to have the same size because they subtend the same visual angle $\alpha_1$ at the eye.
In Diagram 3 the objects are the same size but object $O_4$ appears to be larger than object $O_3$ because object $O_4$ subtends a larger visual angle $\alpha_4$ than the visual angle $\alpha_3$ subtended by object $O_3$.
For small visual angles the angular magnification or magnifying power of an optical instrument is equal to  
$\dfrac{\text{visual angle subtended by the image when looking through the telescope}}{\text{visual angle subtended by the object at infinity when viewed with the naked eye}}$
When looked at with the naked eye the Moon has an angular diameter of about $0.5^\circ$.
When observed with a telescope with a magnifying power of $10\times$ it has an angular diameter of $5^\circ$.
It appears to be ten times larger.
Now with a microscope you first have to consider when an object looks bigger if observed with the naked eye.
The closer the object is to the eye the bigger is the visual angle but there is a limit.
The closest position that you can bring the object up to the eye with the object still in focus is called the near point.
For the “average” eye the distance of the near point from the eye, the least distance of distinct vision, is approximately 25 cm.  
Again for small angles, the angular magnification or magnifying power of a microscope is defined as 
$\dfrac{\text{visual angle subtended by the image when looking through the microscope}}{\text{visual angle subtended at the eye by the object when it is at the near point of the eye}}$
When the objects and images are at distances which are comparable to the focal length of the lens or mirror the linear magnification and the angular magnification or magnifying power are the same. 
It is when the distances are very large that one can only use angular magnification or magnifying power.
A: When you peer into the eyepiece of an optical instrument, you see a virtual image.
I don't know about any difference between "magnification" and "magnifying power", but what I would call "magnification" is the ratio between the apparent visual size of the virtual image of some object, and the visual size of the same object if you observed it from the same distance without using the instrument.
Note: visual size is not a measure of length, but a measure of angle.  It measures how much of your field of view the image fills.
