I'm assuming you mean DNA sequencing (excluding things like RNA-seq).
Is Sanger sequencing the first generation?
From Metzker 2010:
The automated Sanger method is considered as a "first-generation" technology
Is second generation sequencing the same as next generation sequencing?
From the same source:
and newer methods are referred to as next-generation sequencing (NGS).
The "newer methods" in the review are:
- Emulsion PCR based like Roche/454
- Solid phase amplification like Illumina/Solexa
- Single molecule like Helicos BioSciences (bankrupt in 2012) and Pacific Biosciences (commercially available)
As of today, is there any commercial 3rd generation sequencing technology in use (or is it still in development)?
The development of single molecule methods (third item in the above list) has historically taken longer than the other two, so sometimes they are referred to as third generation. Here's another paper referring to PacBio as third generation.
A recent paper from PacBio itself makes the distinction of halting: Second generation is "attach nucleotide, scan, cleave off dye, wash, repeat" while third generation has the polymerase proceed without stopping while scanning occurs concurrently.
Ultimately, I think the "generations" are more useful to make the "Sanger vs. Not Sanger" distinction. Past that, the number of a generation is more of a marketing term.