Incoming links: Bridge amplification
The by far dominating DNA sequencing company of the late 2000's and 2010's due to having the smallest cost per base pair.
To understand how Illumina's technology works basically, watch this video: Video 1. "Illumina Sequencing by Synthesis by Illumina (2016)".
The key innovation of this method is the Bridge amplification step, which produces a large amount of identical DNA strands.