Hurkyl's got it basically right.
Maybe waxing technical will help eliminate confusion, so... permit me :)
A theory is just a set of sentences of first-order logic (FOL) in some language. Examples include: the set of all sentences (which has no models), the theory of Abelian groups, ordered fields, ZFC, Peano arithmetic (PA), the empty set of sentences. If T is a theory and p a sentence in a given language, we write
T |- p
if p can be deduced from T in FOL -- T proves p. "|-" is intended to be one symbol ('turnstile'). If T is empty, then one drops the 'T' before the turnstile. In this case,
|- p
means p is provable, outright, in FOL.
Godel's Completeness Theorem states that for all T and p,
T |- p iff p is true in all models of T
(= every model of T is a model of p -- models whose structure is appropriate for the language of T and p, that is).
By the Completeness Theorem,
|- p iff p is true in all models, i.e. p is universally valid.
Valid sentences include the tautologies -- sentences true solely by virtue of their propositional form, such as (p or ~p), where '~' is negation. (As an example of a valid but non-tautologous sentence schema, consider Ex Ay P(x, y) => Ay Ex P(x, y).)
A theory is called complete if for every p,
either T |- p or T |- ~p
So, if a theory T is not complete (incomplete), then for some sentence p,
neither T |- p nor T |- ~p
Of course it's true for any p that
|- (p or ~p),
hence for every theory T,
T |- (p or ~p);
but only some theories are complete! The empty theory is certainly not complete: not every sentence is either universally valid or inconsistent (true in no models, just plain false). The theory of Abelian groups is not complete; PA isn't; ZF(C) isn't; the theory of real closed fields is; Presburger arithmetic is.
A theory T is consistent if no contradiction can be derived/deduced from T (= not every sentence can be deduced from T). An equivalent formulation of the Completeness Theorem is:
T is consistent iff T has a model.
It's easy to see that the Completeness Theorem implies this; to prove the converse, the Deduction Theorem,
T |- (p => q) iff T + {p} |- q
comes in handy (using '+' for union).
Again by the completeness theorem, if both T + {p} and T + {~p} have models, then T proves neither p nor ~p (p is independent of T) -- and conversely. Because both ZFC + {CH} and ZFC + {~CH} have models, both are consistent, and ZFC proves neither CH nor ~CH -- CH is independent of ZFC. Nevertheless,
ZFC |- (CH or ~CH)
since of course
|- (CH or ~CH).
I've managed to write all of the above without saying much at all about truth of sentences in models. Let me change that :) in order to note a few final relevant facts. If M is a model, considered as a structure, let T(M), the theory of M, be the set of all sentences of the appropriate language which are true in M. T(M) is always a complete theory. Nevertheless, it is certainly possible to have T(M) = T(M') for non-isomorphic M and M' -- indeed, M, M' can have different, and then necessarily infinite, cardinalities (by the Lowenheim-Skolem-Tarski Theorem). First order theories -- even complete theories -- can characterize only finite models up to isomorphism. Finally, by the Compactness Theorem, if a theory has models of unboundedly large finite sizes, then the theory also has infinite models.
Hope that helps!
I suggested you look at something like the arithmetic of an abelian group. That's certainly an incomplete theory! E.g. \exists x \exists y: x \neq y is independent of the abelian group axioms.
