What does the MAF/MinorAlleleCount mean?
[W]hat is the Major allele [frequency]?


If the Minor Allele Frequency is $p$, then, for a bi-allelic locus, the major allele frequency is obviously $1-p$. 


  [W]hy reporting the second most frequent allele is helpful?


Most polymorphic loci of interest are bi-allelic. Hence, the MAF is an indication of genetic diversity. One could similarly report the expected heterozygosity instead ($2p(1-p)$).

It is common to consider only loci with a MAF greater than a certain value. There may be different reasons for that. Here are possible two reasons.

Sequencing errors

When sequencing your sample, errors are being made. Sure, error rate is pretty good (see this table from Molecular Ecology) but error still do exist. A polymorphic locus with a minor allele count of less than, says, 3 alleles might well just be caused by sequencing errors.

To avoid considering as polymorphic loci that are in reality not polymorphic, we often just disregard any data with MAF is below a given threshold.

$F_{ST}$

$F_{ST}$ is a measure of genetic differentiation among population. When computed on loci with low MAF, the statistic is biased (Beaumont and Nichols, 1996). Hence, it is usual to only consider loci with a MAF greater than an arbitrary threshold before computing $F_{ST}$