The R4 is a cartridge that you insert a microSD card into (up to 4GB at the moment for the regular R4) that is the same size as a regular DS cart.
I use DVDFab Platinum to convert shows and movies to a .DPG file that can be played on the DS. There are other ways I'm sure.
It comes with a CD that has the files you need to put on the microSD card that contain the apps to play video, pictures and ROMs.
When you turn the unit on, the touch screen has three options: one button for games, another for multimedia (video, pics), and a third for booting from Slot 2 (the larger slot on the front for SP games. You can also change the brightness, and select whether you want it to enable a soft reboot (hold down all the buttons to reset the DS)
It's very customizable, for example you can change the menu backgrounds, etc. Mine is a pic of my daughter holding the DS. Very cute!
If you only have the R4, you can only play DS roms (games). To play GBA/SP games, you need a Slot 2 unit. They go for around $20 online. (I don't need one since I only play DS games)
The cost of anR4 depends on where you get it and how long you want to wait. My first was from Dealextreme, who no longer sells them and was around $40 shipped, but took more than a month to get. My second was almost $60 with shipping, but only took 4 days.
Let us know if you have any more questions!