Let me work in homology, which is closer to the way someone like Lefschetz would have thought about it. Given a smooth complex projective variety in $X\subset \mathbb{P}^N$, we would like to understand the homology inductively. So take a general hyperplane $H\subset \mathbb{P}^N$,
then $Y=X\cap H$ is again smooth (Bertini) of smaller dimension. The Lefschetz hyperplane theorem says that $H_i(Y)\to H_i(X)$ is surjective when $i<\dim X$. So in this range,
all of the homology is effectively captured by $Y$. But when $i=\dim X$, the simple minded induction breaks down, because there will be a kernel which would be the *primitive homology* in the middle degree. This is the part that is genuinely new, and that needs to be understood on its own terms. This perhaps the simplest answer. 

Switching to cohomology,  the hard Lefschetz theorem shows that cohomology decomposes into primitive parts, so these are the essential constituents for cohomology. The hard Lefschetz theorem has an enormous number of implications for the topology of smooth projective varieties
or more generally compact Kahler manifolds. Dan's answer gives one such application. Here is another: the Betti numbers of smooth projective variety satisfy $b_0\le b_2\le b_4\ldots$
and $b_1\le b_3\le\ldots$ up to the dimension, after which they decrease. **Added in response to the follow up question:**  I don't have the time or energy to create a new answer, so I'll comment here. The successive differences $b_3-b_1$ etc. give the dimensions of primitive cohomology.