- Problem A is in NP if:
B is in NP and you can reduce B to A in polynomial time -> B $\leq_p$ A
Not quite right. The class $P$ is a subset of $NP$ anyway, and hence A is already $NP$ if it is in $P$. The question is whether A is in P or not.
By reducing B to A in polynomial time, you prove that any polynomial solution of A is a polynomial solution of B. Assuming B is in $NP$ gives no useful information about A, since it is possible that B is in P as well (remember that $P \subseteq NP$. However, assuming (and probably that is what you meant), that B is NP-Hard, then, by reducing B to A, we prove that A is also hard for NP class, and hence unless P=NP, the problem A is not in P.
Now back to the question. What we know so far, $P \subseteq NP$. There are more classes above $NP$ like $EXP$ and under $P$ like $L$ for example. Which means,
$$L \subseteq P \subseteq NP \subseteq EXP.$$
Check the link for more information about the classes. However, one characterization of problems in NP, is that they all admit a certifier, that can in polynomial time in the size of the input, say if the given instance is a yes instance. For example, for travelling salesman, which is an NP-Complete problem, the certificate is a path. You can clearly certify it, by checking if it visits all cities and each city exactly once and return to the beginning. Since $P \subseteq NP$ the same works for $P$ problems. Like shortest path. given a shortest path between two vertices, you can certify it it is a shortest path. For example, the certifier can compute a shortest path in polynomial time and compare the lengths.
Now the interesting part about certificates, is to certify if a problem admits a certificate for the negative answer. That means, given an instance and a certificate, the certifier can tell in polynomial time, if the instance is not in the language. Problems that admits such a certifier belong to the class $co-NP$. It is believed that $NP \neq co-NP$ unless $P = NP$. The class $P$ belongs to the intersection of both classes and hence we can certify both negative and positive answers in polynomial time.
Summary
Reducing $A$ to a polynomial problem proves that it is solvable in polynomial time.
Reducing an $NP$-Hard problem to $A$, proves that $A$ is NP-Hard and probably not in $P$. In this case, even if the problem is in $NP$ and hence is $NP$-complete, we can only say about $A$ that it is $NP$-hard, we still do not not if it is in $NP$.
A problem is in $NP$ if there is for each instance that belongs to language of the problem, a certificate, that can be certified in polynomial time in the size of the instance.
A problem is in $co-NP$ if there is for each instance, that odes not belong to the language of the problem, a certificate, that can be certified in polynomial time in the size of the instance.
The class $P$ belongs to the intersection of $P \cap co-NP$.
