What is "contigs" in Picard's ReorderSAM?
I'm not familiar with picard and their reorderSam function, but as far as I know/understand from their documentation they mean this:
The ordering of the contigs while using a reference sequence.
Like this: 


   Figure 5: Anatomy of
  whole-genome assembly. In whole-genome assembly, the BAC fragments
  (red line segments) and the reads from five individuals (black line
  segments) are combined to produce a contig and a consensus sequence
  (green line). The contigs are connected into scaffolds, shown in red,
  by pairing end sequences, which are also called mates. If there is a
  gap between consecutive contigs, it has a known size. Next, the
  scaffolds are mapped to the genome (gray line) using sequence tagged
  site (STS) information, represented by blue stars. © 2001 American
  Association for the Advancement of Science Venter, C. et al. The
  sequence of the human genome. Science 291, 1304–1351 (2001). All
  rights reserved. (source)


ReorderSAM (Picard)
So in Picard you have your INPUT (File), the reads in this file are then mapped on the REFERENCE (File). This can also be seen in their code: 

// write the reads in contig order
109
            for (final SAMSequenceRecord contig : refDict.getSequences() ) {
110
                final SAMRecordIterator it = in.query(contig.getSequenceName(), 0, 0, false);
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                writeReads(out, it, newOrder, contig.getSequenceName());
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            } 


(code source)


  ReorderSam reorders reads in a SAM/BAM file to match the contig
  ordering in a provided reference file


Some more background
There are two main approches two obtain a genome sequence: 


  there are two "main" approches for this:  g. Second-generation
  sequencing technologies produce millions of short(a few hundred bp)
  strings of nucleotides (reads), which is ideal for resequencing when
  reads are mapped to a reference genome (reference-based assembly). De
  novo genome assembly based on second-generation sequencing is
  challenging due to difficulties with GC- or AT-rich and homonucleotide
  DNA stretches, which are under-represented in the sequencing output (source)


The characteristics of these are: 
de novo


  
  no bias towards a reference genome 
  no template to adapt to 
  the    assembly is normally more fragmented 
  it normally works better for large-scale/median scale differences (source)
  



reference mapping


  
  less contigs
  in most methods the reads that don't map are not used in the final sequence (this is also the case with reorderSAM: Reads mapped to contigs absent in the new reference are dropped
  you look what is similar to your reference genome
  SNPs and very small veriations are more easily positioned and compared among groups (source)
  


I would highly recommend to watch this short animation to differentiate between these two and understand what reference genome mapping is.