What is the difference between double layer and dual format DVD
drives?  I am going to buy an external DVD player today and this
paradigm is foreign to my little brain.

Hello dexterelectric-ga,

Thank-you for your question.

A dual format DVD drives allows you to read/write two different
formats of DVD called +R and -R.

It is preferable to have a dual format drive/player as this allows you
to play twice as many DVD's than a single format drive/player (as they
play both +R and -R DVD's).  For instance, say your friend made you a
DVD in the -R format for you to look at but you had bought a
single-format DVD+R drive you may not be able to access the data on
the disk.  If you had bought the dual format player there would be no
problem in accessing the data as your player reads both +R and -R.

"DVD discs come in many different formats. There are five major
formats on the market today: DVD-R, DVD-RW, DVD+R, DVD+RW, and
DVD-RAM. If a device can read more than one type of DVD disc format,
then it is referred to as a dual-format or all-format drive. DVD-R,
DVD-RW, DVD+R, and DVD+RW disc are the most common and are mainly used
to create video and data discs."
https://iomega-na-en.custhelp.com/cgi-bin/iomega_na_en.cfg/php/enduser/std_adp.php?p_faqid=17623

The different formats of DVD are discussed here:
http://en.wikipedia.org/wiki/DVD-R
http://en.wikipedia.org/wiki/DVD%2BR

A double (or dual) layer DVD is a DVD that allows you to store twice
as much information on one disk by recording the data on two different
"levels" rather than a single "level" as is usual.  Double layer DVD's
are usually more difficult to find to buy and are usually more
expensive (often more than twice as much!).

"Dual Layer recording allows DVD-R and DVD+R discs to store
significantly more data, up to 8.5 Gigabytes per side, compared to 4.7
Gigabytes for single-layer discs."
http://en.wikipedia.org/wiki/Dual_layer

"A dual-layer disc has two layers of data, one of them
semi-transparent so that the laser can focus through it and read the
second layer. Since both layers are read from the same side, a
dual-layer disc can hold almost twice as much as a single-layer disc,
typically 4 hours of video"
http://www.dvddemystified.com/dvdfaq.html#1.18

"Dual Layer disks are DVD?s that have multiple layers. These layers
can contain over four hours of information on a single side of the
disk. All this means is that more information can be held on a dual
layer DVD so that you can get more out of the DVD."
http://www.dvdmagnet.com/help/explanations.htm

There is a full FAQ (Frequently Asked Question) file on DVD's here:
http://www.dvddemystified.com/dvdfaq.html

There is an excellent guide to how DVD's work here:
http://electronics.howstuffworks.com/dvd.htm/printable

I hope this information is sufficient to answer your questions but
should you require any further assistance please do not hesitate to
ask for clarification.