The "constants" of a structure (model theory) A structure is composed of the domain, constants, relation symbols, and function symbols.  I understand all of the ingredients of a structure immediately except for the constants.  The definition in my book (A Shorter Model Theory) is

A set of elements of $A$ called constant elements, each of which is named by one or more constants.  If $c$ is a constant, we write $c^A$ for the constant element named by $c$.

An example of a structure with constants is 
$$\langle \mathbb{R}, +, -, \cdot, 0,1, \le \rangle$$
I understand that $0$ and $1$ are special because they are the identity elements for addition and multiplication, respectively, but why are they the constants, and in what sense do we need to specify them?  I understand that I get a totally different structure if I where to use $<$ instead of $\le$, but I don't see what is so special about the constants.  Why not have a structure of the following type:
$$\langle \mathbb{R}, +, -, \cdot, \frac{1}{2},\pi, \le \rangle$$
Is this different and if so, in what sense?  Viewed as a group $\mathbb{R}$ has many choices of elements but only one identity element so I can make sense of specifying the identity element when talking about a group, but in what sense are the symbols related when we write out the structure (or signature)?

One good reason to specify constants in a signature for models is that you want to make sure that these are preserved by homomorphisms. Consider, as in your example, the signature of ordered fields $\{ +, -, \cdot, 0,1, \leq \}$. A homomorphism $f:A \to B$ of structures of this signature will always satisfy $f(0_A) = 0_B$, and similar for $1$, whereas this need not be the case for the signature $\{ +, -, \cdot, \leq \}$. 
Moreover, you can refer to constants in formulas that are then quantifier-free. For example, consider a field $K$. Within the first signature, the formula $\varphi(x) = (x = 0)$ is quantifier-free, whereas it is not possible to express the same statement in the second signature without the use of quantifiers. This can be quite important when proving properties of certain classes of models.
