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In general sense, convariant and contravariant isn't a interesting thing. But what confuses here that is the meaning of those words. 

Actually, if we think of two different basic vectors than a basis vector will be covariant another will be contravariant. But a basis vector must be smaller than another basis vector. Let we have a vector which looks like this : $\vec{A}=2\vec{e_1}+2\vec{e_2}$. Here $\vec{e_1}$ and $\vec{e_2}$ is basis vectors. I am taking two e_1 and e_2 to make a vector line. Now I am going to decrease length of those basis vectors let $\tilde{\vec{e_1}}=\frac{1}{2}\vec{e_1}$ and $\tilde{\vec{e_2}}=\frac{1}{2}\vec{e_2}$. To make that $\vec{A}$ using these new basis vectors I must increase coefficient of $\vec{e_1}$ and $\vec{e_2}$. Systematically, that vector should be $\vec{A}=4\tilde{\vec{e_1}}+4\tilde{\vec{e_2}}$. So that I can find the same vector this way. But here we can write $\tilde{\vec{e_1}}=e^1$ (It's superscript not exponent). And $e_1$ is called covariant and $e^1$ is called contravariant. 

We usually use the method to define spacetime from different parts. There's possible way to transform them also (there's a Wikipedia article which talks about it I will add it later). 

When using contravariant and covariant in a single term that represent that, we are trying to represent a vector using different basis. Like as, we are going to define plain line respectively through x,y and z axis $e_1$, $e_2$ and $e_3$. Now I am going to transform them rather than decreasing their length and I am call those transformed basis vector as $e^1$, $e^2$ and $e^3$. To find a curvy vector we must use those transformed and normal basis vectors together (we can call that vector by normal basis vectors but it will be easier to find that vector by mixing all basis vectors) $\vec{A}=g^1_2+g^3_1$. For finding any kind of vector we use i,j,k respectively for 3 dimensional coordinate. We can take another vector $\vec{B}=\sum_{ijk} g_i^{jk}+g_k^{ij}$. But Einstein said that we are summing them every time so we can remove the summation expression from that equation but that doesn't mean we aren't summing we are summing by default. So the same vector will look like, $\vec{B}=g_i^{jk}+g_k^{ij}$.