In a polytope, what are the difference and relation between facet and face? How are they defined respectively? Thanks and regards!
Usually facet is synonymous with maximal face. Or in other words, if the polytope is of dimension $d$, the facets are the faces of dimension $d-1$, or codimension $1$.
A face is just a common name for $\emptyset$, vertices, edges, and so on. Often one says that a $k$-dimensional face is called an "$n$-face". Usually one also says that the whole polytope is a face also (this is to ensure that intersection of faces is also a face). The mathematical definition of a face varies in the literature (as the Wikipedia article mentions) - but often one says that a face of a polytope is a subset of the polytope maximizing some linear functional (though this definition is not very intuitive...)