Sorry to dig up such an old question, but I wanted to mention some more applications of Model theory that has perhaps been not covered in the other answers:

1) Using the basic algebraic fact that given a field $\mathbb F$ and finitely many polynomials $f_1,...,f_n \in \mathbb F[X]$, there is a field extension $\mathbb K_{|\mathbb F}$ in which all the $f_i$ s has a root; one can show by using the Compactness theorem , that given a field $\mathbb F$, there is a field extension $\mathbb L_{|\mathbb F}$ in which every polynomial $f \in \mathbb F[X]$ has a root. This in turn can be used iteratively to build an algebraically closed field extension $\mathbb K_{|\mathbb F}$ . Then taking the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$, we get an algebraic extension of $\mathbb F$ which is algebraically closed i.e. we get an algebraic closure of $\mathbb F$. The Compactness theorem can also be used to prove the uniqueness of algebraic closure.

2) The first order theory of Real Closed fields in the language of ordered fields has quantifier elimination. This can be easily used to prove certain facts about " semi-algebraic sets" (i.e. sets which are Boolean combination i.e. finite union , intersection and compliment of sets of the form $\{\bar x : p(\bar x) =0\}$ and $\{\bar x : p(\bar x) <0\}$, where $<$ is the canonical order on a real closed field and $p$ is a multivariate polynomial with coefficients in the field)

(i) If $\mathbb F$ is a real closed field and $f: \mathbb F^m \to \mathbb F^n$ is a semi-algebraic function (the graph of the function is a semi-algebraic set) , then $f(A)$ and $f^{-1}(B)$ are semi-algebraic for any semi-algebraic sets $A \subseteq \mathbb F^m , B\subseteq \mathbb F^n$. This is often called the generalized Tarski-Seidenberg theorem.

(ii) Every non-negative element in a real closed field is the square of a unique non-negative element; this defines the square root of a non-negative element. For a real closed field $\mathbb F$, using the $\mathbb F$-valued norm $|(x_1,...,x_n)|:=\sqrt {x_1^2+...+x_n^2}$ on $\mathbb F^n$ , we get a topology on $\mathbb F^n$. Using quantifier elimination, it can be easily shown that the closure (under the above mentioned topology) of any semi-algebraic set is again semi-algebraic.

(iii) the theory of RCF has quantifier elimination and has a prime model, hence it is a complete theory. Using this it can be shown that If $\mathbb F$ is a Real closed field and $f : \mathbb F^m \to \mathbb F^n $ is a continuous (w.r.t. the topology ) semi-algebraic function, then $f$ carries closed and bounded sets to closed and bounded sets (bounded in the sense of the $\mathbb F$-valued norm).

(iv) Using the completeness of RCF and quantifier elimination , it can also be shown that if $f: \mathbb F \to \mathbb F$ is semi-algebraic , then for any open-interval $U\subseteq \mathbb F$, there is $x \in U$ such that $f$ is continuous at $x$.

And there are many more similar results.