what is Prime Gaps relationship with number 6?

To provide a different perspective on Vhailor's answer: note that if $p$ is a prime $\gt 3$, then $p+6k$ is guaranteed not to be divisible by $2$ or $3$ for any $k$; in effect these gaps are 'pre-sieved' to weed out possible multiples of $2$ and $3$ that could keep the number at the other end from being prime. If you expanded your chart out further you would see similar spikes at the multiples of $30$, since those numbers are also 'pre-sieved' for $5$. (In fact, if you were to expand your table out to all the prims less than $2\times 10^{35}$, you would find the total number of gaps of length $30$ to be more than the number of gaps of length $6$ - see http://mac6.ma.psu.edu/primes/ for the details!)

All prime numbers except 2 and 3 are of the form $6k±1$, so whenever you fall on a pair $6k+1$, $6l+1$ their difference will be a multiple of $6$, same goes for a pair $6k-1$,$6l-1$.

http://en.wikipedia.org/wiki/Primality_test#Naive_methods

Excepting the first two gaps, all prime gaps are between numbers that are either $1$ or $5$ modulo $6$. Under the assumption that both cases are equally likely, half the prime gaps will be between numbers in the same class, and therefore of size $0$ modulo $6$, and the other half will be between numbers in different classes, which split up into sizes that are $2$ and $4$ modulo $6$. Since each of the latter cases only gets one quarter of the total, it is clear that ignoring all other factors, gaps that are $2$ or $4$ modulo $6$ are about half as likely to occur as gaps of the same approximate magnitude that are $0$ modulo $6$. You can check this in your chart. (Particular gap sizes are also subject to influences of other primes than $2$ or $3$, which explains some other irregularities one can observe.)