You will get different answers form different people.
"Second and third generation" and the terms so on refer to DVD-Video players as well as DVD-ROM drives. Generally speaking, they simply mean the newer versions of DVD playback devices, but not the DVD products to record, play video games etc.
In some people's opinions, the second generation DVD players came out in the autumn of 1997, and the third generation refers to those came out in early 1998. Meanwhile, according to others, the second generation DVD players are HD players came until 2003.
Many conflicting variations occur between these extremes, including the viewpoint that DTS-compatible players or Divx players or progressive-scan players or 10-bit video players or players that can play The Matrix constitute the second, third, or fourth generation.
This thing is a little clearer in the PC world. The second generation (DVD II) usually means 2x DVD-ROM drives that can read CD-Rs, while the third generation (DVD III) usually means 5x (or sometimes 2x or 4.8x or 6x) DVD-ROM drives.