The key to it all is understanding the difference between what is disc media, and what is encoding ("encoding" is often informally referred to as the format of the disc).
Haven't I seen this "AVCHD" before?
AVCHD was created by Sony and Panasonic but is also supported by other camera manufacturers such as Canon and JVC. There is also a range of professional camcorders labelled AVCCAM.
Long ago the re-writeable "-RW" discs were popular for a short time, but flash drives (aka USB drives, thumb drives) quickly became cheaper and offered larger storage, making re-writeable optical discs irrelevant for saving data. They might still be of use if you want to write films to the discs and write over them again a number of times.
Alternatively the disc can be encoded to play video or audio in a specific format (the most common being audio on a CD and video on a DVD, but there are others). Encoding video to play back on a domestic DVD player, for example, needs to be formatted in a certain fashion. Simply burning video files onto a disc is no guarantee it will play in a CD player or DVD player, so software needs to encode the video/audio, such as iTunes or Nero for burning audio CDs, and iDVD or PowerDVD for burning video DVDs. These encoding schemes are referred to by names such as DVD-Video, DVD-ROM, CD-Audio and CD-ROM ("ROM" refers to "read-only memory"). If a disc is marked "DVD-Video" then you know it is encoded to play back on a domestic DVD player for viewing on a TV (and could also be played on a computer if you have the right software).
Note: Just to confuse things, some fancy DVD players have extra capabilities and can play certain media from CD-ROMs and DVD-ROMs, such as MP3 audio files and AVI video files. There was also a short-lived encoding scheme named Video-CD or VCD, which can be played by some DVD players.
Encoding AVCHDSo the AVCHD disc is a new way of encoding onto existing disc media. All the benefits of watching high definition films but using the regular DVD burner in your computer. Video files are encoded to a certain codec (similar to the codec used by Blu-ray discs). Although it is burnt to a regular DVD disc, the video can't be played by a regular DVD player, since it's codec is outside the DVD-Video specification. Many domestic Blu-ray players can play AVCHD discs, possibly due to the similarities in the codecs between the AVCHD-encoded videos and the Blu-ray-encoded videos.
Burning an AVCHD disc is possible using a variety of software. Certain versions of Adobe Premiere and Final Cut Pro have the capability, and there is specific disc-burning software such as Roxio Toast. These applications will encode your video to the correct format and burn the disc media.
Some superfluous info: Encoding a disc to play a film is performed by converting the movie files using a codec, and each encoding scheme has a strict specification. For example, DVD-Video is encoded with the MPEG-2 codec, and a Blu-ray Video is encoded with either H.264 or MPEG-2 codecs.
Now because AVCHD videos are file based, and burning them to a DVD disc is a way of storing the files, it is possible to do the same with a memory card. It is not advisable to pop your memory card onto your Blu-ray player's disc tray (some crunching noises may result) but many Blu-ray players have SD card slots, or perhaps a USB socket where a memory card reader can be plugged in. If you want to play unedited shots you can usually plug in your AVCHD camcorder and the Blu-ray player will play the files.
NomenclatureIt is common to get mixed up when referring to different encodes and media. To refer to "DVD" could mean the disc media, or the DVD-Video encode that allows the disc to play in a domestic DVD player. But then what is a "DVD player"? It could be the set-top box that plugs into your TV and allows you to watch films, or it could be the optical drive in your computer. But the DVD drive in your computer won't be able to play films if the computer doesn't run the right software. So it can be a bit of a minefield if you're particular about getting your terms absolutely correct.
As a user of both Windows and Mac for years, I find it quite impressive how Mac computers are so well prepared for using optical media. The Mac OS X system includes licences and software for playing and burning films onto DVDs, easy burning of files onto a DVD-ROM and burning custom iTunes playlists onto CDs. If you're using Windows there is third-party software available (and some are pre-installed on new Windows computers) such as PowerDVD for playback of DVD-Video discs, and Nero for burning CDs. A Mac user usually has everything to hand whereas a Windows user might need to hunt around bit.
Newer Mac computers such as the MacBook Air and the latest Mac Mini have abandoned optical drives as the world moves to flash storage and online software distribution. Fortunately there are plenty of USB-connected optical drives available for burning and playback of CDs, DVDs and now Blu-rays.