Some are quite different than others:
Axiom: This is what you are taking to be the ground truth. For instance, Peano axioms axiomatize natural numbers, and you can use Dedekind cuts to axiomatize reals (You can read Dedekind's cut and axioms for more info)
You can use this somewhat interchangably with the word "definition". (You can say this is how you "define" natural or real numbers), but generally a definition goes as:
"Defintion: We say a natural number is BOO if it satisfies the following axioms:
1) If you add one to any number which is BOO, the resulting number is not BOO.
2) If you add one to any number which is not BOO, the resulting number is BOO.
3) 1 is not BOO.
Now, let's make a proposition. (Generally, this just means a statement that can be either true or false. It is something that they propose; this is I think mostly used if they are not going to prove it for a while.)
Proposition: "Every BOO number is even."
Theorem: This is essentially a mathematical truth; anyone claiming one of these better give you a proof of it. Since we will prove the above proposition, let's rewrite it as:
Theorem: "Every BOO number is even."
Now, to help us prove it, we are going to prove two mini-theorems, more commonly referred to as lemmas.
Lemma 1: "2 is BOO"
Proof: 1 is not BOO by axiom 3, so by axiom 2, 1+1 = 2 is BOO.
Lemma 2: "If n is BOO, n+2 is BOO"
Proof: If n is BOO, by axiom 1, n+1 is not in BOO, but then by axiom 2, (n+1)+1 = n+2 is in BOO.
Now, the proof of the theorem would be to show that every number of the form 2 + 2 + 2 + ... + 2 would be divisible by 2. You can use the fact that multiplication by 1/2 distributes over summation, but that would require perhaps a lot more definitions than what you want to use. You can also use something called induction, which is a proof method. It depends on how you want to define even-ness.
Now, let's write a corollary from our theorem:
Corollary: 2 is even. (We know this follows from our theorem, since we proved earlier that every BOO number is even and we know that 2 is BOO.)
Law: It refers to big observations, you can write a "law of BOOs as: Every number is either BOO or BOO + 1." Laws are not that common, but we have a few big ones: Law of large numbers, and De Morgan's laws. You can imagine them to be very big or useful theorems. (There are a lot of big and useful theorems like central limit theorem that are not called laws).
Hope that answers it with a toy example!