In this paper, the authors consider a nonlinear differential-algebraic equation (DAE) of the form \[ E(x)\dot x=F(x),\tag{1} \] where \(x\in X\) is a vector of the generalized states and \(X\) is an open subset of \(\mathbb R^n\) or an \(n\)-dimensional manifold. Two special cases of the DAE (1), the semi-explicit DAE and the linear one, are also considered. DAEs play an important role in modeling practical systems such as electrical circuits, chemical processes, mechanical systems, etc. In this paper, a geometric analysis of (1) is given using some tools from nonlinear control theory. There are three main results of this paper. The first result concerns analyzing a DAE everywhere (i.e., externally) or considering the restriction of the DAE to a submanifold (i.e., internally), which leads to the notions of external equivalence and internal equivalence, respectively. In order to analyze the existence of solutions, the authors construct the so-called locally maximal invariant submanifold (a manifold where the solutions of a DAE exist) via a geometric reduction method. This is a revised version of the geometric reduction method that was used for DAE control systems. A practical implementation of this method is also described via an algorithm. As the second result, the authors show that for any nonlinear DAE, by introducing the so-called driving variables, one can associate a class of nonlinear control systems. Furthermore, the driving variables in this explicitation procedure can be fully reduced under some involutivity conditions which explains when a nonlinear DAE is equivalent to a semi-explicit one. The last result is to use concepts of the nonlinear control theory such as zero dynamics, relative degree and invariant distributions to derive two nonlinear generalizations of the Weierstrass form which is well known in the case of regular linear DAEs. Some examples are also given for illustrating the main results.