Can someone explain in a simple way what "reducible" mean in complexity theory? I find the word "reducible" used in complexity theory not very intuitive, and too general taken on a face value.

What does it exactly mean by problem A reducible to B? 


Does it mean that A can be reworded into B? 
i.e. Finding Vertex Set is actually the same as finding an independent set?
Does it mean that an algorithm that solves problem B will solve A? 
i.e. An algorithm that finds longest path will in fact also find the shortest path?
Does it mean that A can be proved to be in the same complexity class
as B? 
i.e. A is NP-Hard, B is NPC, then if A is reducible to B, A is in fact NPC?


In other words, A is reducible to B, what can we say about A or B?

As you can see my understanding is quite flaw at the moment, can someone please clarify the application of the word for me maybe even rephrase it into something more intuitive?
The most common notion of reducibility is many-one polynomial time reducibility.

A problem $A$ is many-one polynomial time reducible to a problem $B$ if there is a polynomial time function $f$ such that $x \in A$ iff $f(x) \in B$.

This definition implies that if $B$ is solvable in polynomial time, then so is $A$.

Other notions also exist, but at this stage you're probably better off concentrating just on this one, since it is the one used to define NP-completeness.