Element formulations in finite element analysis may follow different approaches. A classification of elements in equilibrium and compatibility is presented. Analysis and comparison of some classic plane elements show that, non-conforming, conforming and super conforming elements can be constructed based on the compatibility theory, and analytical trial functions can be employed to help the construction of equilibrium elements. The researches of compatibility focus on the boundary conforming between elements, which conform bases of simplex conforming, non-simplex conforming, non-conforming and super conforming elements. The researches of equilibrium focus on the equilibrium principle in element or between elements, which emphasize high-precise completeness of analytical trial functions and weight functions. The benchmark tests show that although all kinds of elements have their own advantages and shortcomings, elements that satisfy both the equilibrium inside elements and the compatibility between elements deliver generally better performance.