It is a way to test if a given series of non-negative real
numbers converges or not. The technique is to compare it with
another series of non-negative real numbers whose behaviour is
known.
The basic idea is like this:
Suppose that $\sum a_n$ converges, where $\forall
n~~a_n\geq0.$ So we are adding more and more and
more $a_n$'s (so the sum is growing and growing and
growing), but still the sum is not blowing up to infinity. Clearly,
this means that the $a_n$'s are getting smaller and smaller
and smaller pretty fast. Now suppose that I give you another
series $\sum b_n$ where $\forall n~~0\leq b_n\leq a_n.$
It should be intuitively quite obvious that $\sum b_n$ must
also converge. Well, that is indeed true.
This is called comparison test.
Proof:
Consider the sequence of partial sums $\left(\sum_1^n
b_k\right)_n.$ Clearly, this is a non-decreasing sequence
(since $b_n$' are all $\geq 0$). Also this is bouded
from above, since
$$
\forall n~~\left( \sum_1^n b_k \leq \sum_1^n na_k \leq \sum_1^\infty a_k\right).
$$
Hence the sequence must converge, as required.
The second part is just the contrapositive of the first
($A\Rightarrow B$ is the same as $\neg B\Rightarrow \neg
A$).
[QED]
In fact, you do not
even need $b_n\leq a_n$ to hold for all $n$'s. It is
enough if the ineuqlity holds $\forall n\geq N$ for
some $N.$