Let there be $n$ coordinates $q^j$. The treatment of non-holonomic constraints in Ref. 1 is subpar for various reasons, see e.g. this & this related Phys.SE posts.
However we interpret OP's question (v2) as mostly being about counting independent degrees of freedom in constrained systems, and not so much about non-holonomic constraints per se. Therefore, to gain intuition, let us for simplicity just consider $m$ holonomic constraints
$$\tag{A}f_{\alpha}(q)~=~0,$$
where $m\leq n$ (and where we have suppress possible explicit time dependence in the notation). Granted some regularity assumptions, we may in principle solve the $m$ constraints (A) locally so that the coordinates
$$\tag{B}q^j~=~g^j(\xi, \varphi)$$
become functions of $n-m$ independent physical coordinates $\xi^a$, and $m$ coordinates $\varphi^{\alpha}$, in such a way that locally the $n-m$ dimensional constraint surface
$$\tag{C}\{q\in\mathbb{R}^n|f(q)=0\}$$
is parametrized as
$$\tag{D}\{g(\xi, \varphi=0)\in\mathbb{R}^n| \xi\in \mathbb{R}^{n-m}\}.$$
Thus we have at least two equivalent variational formulations:
Reduced formalism: Replace $q^j$ with $g^j(\xi, \varphi=0)$ in the action $S[q]$. Vary the corresponding action $S[\xi]$ wrt. the $n-m$ independent variables $\xi^a$.
Extended formalism: Replace the action $S[q]$ with
$$\tag{E} S[q,\lambda]=S[q]+\int\!dt~\lambda^{\alpha}f_{\alpha}(q).$$
Vary the corresponding action $S[\xi,\lambda]$ wrt. the $n+m$ independent variables $q^j$ and $\lambda^{\alpha}$.
The role of the $m$ Lagrange multipliers $\lambda^{\alpha}$ can be view as putting the $m$ variables $\varphi^{\alpha}=0$, so that only the $n-m$ physical variables $\xi^a$ remains, and formulation (2) reduces to (1).
References:
- H. Goldstein, Classical Mechanics, 3rd ed.; Section 2.4.
Best Answer
If you have a mechanical system with $N$ particles, you'd technically need $n = 3N$ coordinates to describe it completely.
But often it is possible to express one coordinate in terms of others: for example of two points are connected by a rigid rod, their relative distance does not vary. Such a condition of the system can be expressed as an equation that involves only the spatial coordinates $q_i$ of the system and the time $t$, but not on momenta $p_i$ or higher derivatives wrt time. These are called holonomic constraints: $$f(q_i, t) = 0.$$ The cool thing about them is that they reduce the degrees of freedom of the system. If you have $s$ constraints, you end up with $n' = 3N-s < n$ degrees of freedom.
An example of a holonomic constraint can be seen in a mathematical pendulum. The swinging point on the pendulum has two degrees of freedom ($x$ and $y$). The length $l$ of the pendulum is constant, so that we can write the constraint as $$x^2 + y^2 - l^2 = 0.$$ This is an equation that only depends on the coordinates. Furthermore, it does not explicitly depend on time, and is therefore also a scleronomous constraint. With this constraint, the number of degrees of freedom is now 1.
Non-holonomic constraints are basically just all other cases: when the constraints cannot be written as an equation between coordinates (but often as an inequality).
An example of a system with non-holonomic constraints is a particle trapped in a spherical shell. In three spatial dimensions, the particle then has 3 degrees of freedom. The constraint says that the distance of the particle from the center of the sphere is always less than $R$: $$\sqrt{x^2 + y^2 + z^2} < R.$$ We cannot rewrite this to an equality, so this is a non-holonomic, scleronomous constraint.