Eigen values and Eigen vectors
In linear algebra, an eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by 
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.[2] Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
PCA Theory
Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest.
PCA is used in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data’s variation as possible. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The 
From either objective, it can be shown that the principal components are eigenvectors of the data’s covariance matrix.
https://en.wikipedia.org/wiki/Principal_component_analysishttps://royalsocietypublishing.org/doi/10.1098/rsta.2015.0202
WHAT IS PRINCIPAL COMPONENT ANALYSIS?
Principal Component Analysis, or PCA, is a dimensionality-reduction method that is often used to reduce the dimensionality of large data sets, by transforming a large set of variables into a smaller one that still contains most of the information in the large set.
Reducing the number of variables of a data set naturally comes at the expense of accuracy, but the trick in dimensionality reduction is to trade a little accuracy for simplicity. Because smaller data sets are easier to explore and visualize and make analyzing data much easier and faster for machine learning algorithms without extraneous variables to process.
So to sum up, the idea of PCA is simple — reduce the number of variables of a data set, while preserving as much information as possible.
Step-by-step application of PCA
https://builtin.com/data-science/step-step-explanation-principal-component-analysis
Explanation of PCA to layman
In some simple words PCA is explained to a grandma.
https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues