Copy number variants (CNVs) are relatively large physical changes in the genome. This article provides an approachable introduction to the topic.
A large study (19,000 individuals) over several diseases led Wellcome Trust authors to conclude that most common CNVs are well tagged by SNPs and so are indirectly being studied via SNP studies, and they felt that these common CNVs are unlikely to contribute greatly to the genetic basis of common human diseases. [PMID 20360734]
This overview of dbVAR, the NCBI database of structural variations, is a more technical survey. It uses numerical nsv identifiers, which are comparable to either the rs numbers or perhaps the ss###.
The Database of Genomic Variants is an upstream resource based on coordinates, not names.
This article shows some CNVs linked to diseases.
By 2008, 37 human disease conditions have been identified, which are either caused by CNVs or whose relative risk is increased by CNVs (Beckmann et al., 2008; Lee and Scherer, 2010, and references therein).
The above and table 3 both provide the first medically relevant CNV data I've seen,
Identical twins seem to have differences in CNVs. 
James Watson discusses copy number polymorphisms related to autism near minute 34 of this video.
[PMID 18204055] extracts CNVs from microarray data
[PMID 17189289] demonstrates the importance of CNVs relative to SNPs
[PMID 17206864] A High-Resolution Map of Segmental DNA Copy Number Variation in the Mouse Genome.
[PMID 17122084] Genome-wide detection of human copy number variations using high-density DNA oligonucleotide arrays.
PDF Poster Determination of Genomic Copy Number Variation (CNV) using the SNPLex Genotyping System (can't direct link)
SNP and CNV variation correlated with altered activity in almost 900 and 240 genes, respectively. The HapMap has been invaluable in detecting variants involved in many diseases and these results suggest that the CNV index will prove similarly useful.
"The remarkable finding was that there is such little overlap in the genes found by using the two indices," commented Dr Matthew Hurles, also a leader of the project at the Wellcome Trust Sanger Institute. "Only about 10% of the activity variants associated with a CNV were also associated with a SNP.
"This suggests that we must include CNV studies in our searches for genetic variation associated with disease or we will be missing a lot of the important genetic effects."
The results show that at least 10-20% of heritable variation in gene activity is due to CNVs. The team found associations that included previously known examples, such as UGT2B17, which has been associated with Prostate cancer, proving that the new approach works well. 
[PMID 17289997] SNPs and CNVs captured 83.6% and 17.7% of the total detected genetic variation in gene expression, respectively, but the signals from the two types of variation had little overlap.
[PMID 17280605] a paper which appears to link CNVs to Breast Cancer
[PMID 17529978] FCGR3B copy number variation is associated with susceptibility to systemic, but not organ-specific, autoimmunity.
DEFB4 Beta-defensin gene CNV is correlated to both Crohn's disease (if you have too few copies) and psoriasis (if you have too many copies)
[PMID 17921354] software. PennCNV: An integrated hidden Markov model designed for high-resolution copy number variation detection in whole-genome SNP genotyping data
[PMID 17910767] discusses several packages for determining CNVs in microarray data