Let $T$ be a infinite first-order theory T, over the empty signature ($\Sigma =$ {}), as follows: $$T = \{\phi_n : n \in \mathbb{N} \}$$ where $$\phi_n \equiv \exists x_{1},\exists x_{2},\exists x_{3},...,\exists x_{n} (x_{1}\not=x_{2} ∧ x_{1}\not=x_{3} ∧ x_{2}\not=x_{3} ∧ ...).$$
Which means: "there exists at least $n$ elements in the given domain".
Can you prove this FO theory is complete?
If the language has no function symbols, predicate symbols, or constant symbols, then the theory is $\omega$-categorical, and therefore complete.
If the language contains a function symbol, or a predicate symbol, or more than one constant symbol, the theory will be incomplete.