A theorem is a statement proved by a chain of reasoning. The theorem is not self evident. It is derived after considering the results of several logical statements (often including other theorems). A famous example of this is the Pythagorean Theorem, which has nearly 400 proofs.
Laws are statements which are inferred by observation. Laws are not proved. Laws are demonstrated based on repeated observations. It expresses a causal relationship between entities under certain conditions, and is often expressed mathematically. The Laws of Thermodynamics (so often quoted without the mathematical context and misstated for that very reason) are good examples.
A theory is an established and respected explanation of a natural phenomenon, acquired through confirmation of its principles through the scientific method - testing, confirmation, and observation and experimentation.
A fact is a true statement under the set conditions. A thing that is indisputably the case. Facts can generally be reproduced to be verified. A single counter example will immediately demote a fact to a false claim.
A proof in science a successful demonstration of a hypothesis under study using evidence and analysis. It is a misconception in many ways, since science actually can't prove anything for certain. In this sense, "proof" is a simplified way of saying "greatly enhanced confidence".
A rule is an informal axiom that expresses a philosophical point. A rule governs how a subject behaves under certain circumstances. A rule indicates increased chance of a certain outcome when a subject is in a certain state. Newton's "rules" or Occam's Razor are examples.
A Conjecture is a statement not (yet) proven, but which intuitively seems to be true, or for which the author has some more complex reasons to believes to be likely to be true. Some deceivingly simple of those resist centuries of attacks by brightest mathematicians.