I think that you should be able to find an answer for all your questions in a simple google search. Here you have some information:

• Properly base pair: I guess that this term is the same as "properly paired" which means that both mates of a read pair map to the same chromosome, oriented towards each other, and with a sensible insert size. Some information about CIGAR string: http://genome.sph.umich.edu/wiki/SAM
• CIGAR has different operations:
  • Standard: M : match or mismatch,I : insertion, D: deletion
  • Extended: N : skipped bases on reference, S : soft clipping, H : hard clipping, P : padding Also if you see the FAQ of Tablet assembly:

What are the "CIGAR-I" features that Tablet adds to SAM and BAM files?

Tablet automatically adds these features types for column positions where an insertion into the reference has occurred (and has been marked up in the CIGAR formatting of the original SAM or BAM file). If you hover the mouse over the feature in the Features Table, then a tooltip will appear that also informs you how many reads at that position were tagged in this way.

• What is a contig? --> http://en.wikipedia.org/wiki/Contig

Finally here you have the paper of Tablet, where I guess that you'll find more answers to your questions