How Can PCA Be Used in Image Analysis
Imagine you have a set of 10,000 images (32 x 32) of faces.

An intuitive way is to think they have a lot in common.
One step farther would be that if you take one of the faces you could generate it from a Linear Combination of the other 9,999 images.
Sounds reasonable, isn't it?

Let's treat the images as a vector space with dimension $ d = 32 \times 32 = 1024 $.
So all we need is $ 1024 $ independent vectors in order to be able to span any images of size $ 32 x 32 $ in general and faces in specific.  

But, if $ 1024 $ images can span all $ 32 x 32 $ images in the world, can't we do better for a subset of this amazing number of different images?
After all we're only after images of faces, human faces.  

So we want smaller number of vectors which spans face images.
Maybe not perfectly, but at least almost perfectly based on much smaller number than 1024.
Can we do that?

Yes!
Actually many good ways to do so.
The specific way which uses Principal Component Analysis (PCA) is called Eiganface (At least one of them).
This method is the most efficient way to minimize the $ {L}_{2} $ norm of the reconstruction error when there is a limitation of using $ k $ elements of the linear reconstruction.  

If one will analyze the $ {L}_{2} $ norm of the error one would see the best elements are the Eigenvectors of the data set.
Those are the result of the PCA process (Or SVD more generally) and in the context of this method, unsurprisingly, they are called eigen faces.

This general problem is called dictionary learning.
Given a set of data we're learning a dictionary (Set of Atoms) which describe it most efficiently according to some measure of the reconstruction quality.

PCA is one of the most used methods for this kind problems.
The other side of the coin of this problems is the Dimensionality Reduction method where PCA is also heavily used.