In most cases, good audio is the key to good video. Camcorder mikes have issues due to their location. However, being tied to a microphone by a wire drastically limits mobility, and wireless means knowing where the audio will be occurring in advance, and is typically limited to one location. And some cameras don't have an external audio input.
In the "good old days", having an external recorder implied having some complex electromechanical linkage to maintain synchronization between the two devices. However, these days, each device generally has a crystal controlled (very accurate) internal source of timing, comparable in accuracy to a quartz watch. As a result, they can operate independently at record time. They can synchronized once during editing, and there is a reasonable expectation that they will maintain synchronization over a usable period.
This normally means starting all of the video and audio recorders at about the same time, then doing something to make later synchronizing easy (such as a clapboard, or just clapping once in a position visible to the cameras). If recorders are to be stopped and restarted, remember to clap each time.
This approach allows scattering audio recorders wherever audio input might be anticipated, or setting up for multichannel recording. It also allows recording with multiple camcorders even when working alone, with locked off wide shots or area coverage. When all tracks are aligned for editing, editing can resemble a multicamera realtime shoot. But it doesn't have to be that elaborate -- it can be used for a single audio recorder and single camcorder.
As for the recorders, some inexpensive mp3 players with voice recording capability can produce very good results. You need to test the quality and nature of the encoding, as most are very poor (e.g. wav encoding at low bit rates, or noise and internal interference). The old iRiver IFP 7XX and 8XX series (now an eBay item) offered good pickup quality, along with the ability to encode the audio at high bit rates with MP3 compression. This allowed a 256MB MP3 player, "useless" as a player due to its limited recording time, to record high quality audio for several hours! These can typically be purchased for about $30. The best I've found is the Zoom H2 at about $200. Audio from the iRiver has to be uploaded and converted (fast), while with the Zoom its just a matter of moving the SD Flash card (faster).
If you do go wireless, remember you want the receiver to be small and battery operated if you are to remain mobile with your camcorder. Most wireless setups are the opposite, with an AC powered large clunky receiver. Also, if the transmitter has an external input, it can be driven from a mixer with multiple wired mikes. This is great, for example, if you are shooting a band and want the best audio quality (subject remains in a relatively fixed position), along with the freedom to move. Use at least one locked off wide camera for cover between moves, and remember to not stop your mobile camera between shots so sync is maintained during editing.
If you are going with wireless, look for the term "diversity" on the receiver, and go that way if you can afford it. The receiver has two separate RF channels, and whichever is stronger at that instant is the one used. This helps considerably to avoid fades and interference, which can happen over very short distances.
A big thanks to whizkid454 and boya84, who have both given lots of excellent advice and information. I'm very impressed and grateful.