If you want to do some googling, the QC metric used by Illumina is called the "chastity filter". Chastity is a measure of the signal to noise ratio and is defined as the "the ratio of the highest of the four (base type) intensities to the sum of highest two."

The chastity threshold is 0.6 and this threshold is applied to the first ~20 positions in the read, regardless of the read length. (I forget exactly how many positions are currently used, as the chastity filtering algorithm has changed over time) At most one base is allowed to fail to meet the 0.6 ratio threshold...if more than one base is below threshold the read is marked as failing QC. (For paired end reads, if either end of a pair fails to meet the threshold both ends are marked as QC failed.)

The most common source of chastity failure is two (or more) adjacent clusters being so physically close together that their signals cannot be measured independently.

QC Failure= too many Ns in your reads

properly paired = you have used a paired end sequencing. http://www.illumina.com/technology/paired_end_sequencing_assay.ilmn . Both left and right reads have been mapped on the same region of the genome at a distance compatible with the expected mean size of the fragments.

"Properly paired" is from the flags set by whatever aligner you've used. For instance, if you use bwa's 'sampe' alingment, you can specify the maximum insert length with -a. For a pair to be "properly paired" it would need to have both reads mapped to the same sequence within this distance.