Using AverMedia EZCapture with Linux is very easy.

This card use a bt878 chipset, so you havo to build the driver from the kernel source.
I have tested the card with linux kernel 2.4.18 and 2.4.22 and I have verified that this card works fine.
After compiling modules, you can load the module with a simple command:

> modprobe bttv card=13

Remember to specify the option card=13 otherwise you could see only images in black and white.
If you try to execute this command, you have to wait a long time (some minutes!).
To avoid this wast of time every Linux boot you can append the string

options i2c-algo-bit bit_test=1

to your modules.conf file.

You can find some documentation in the /usr/src/linux/Documentation/video4linux/bttv directory.

Obviously you need a specific software to capture video and save images and movies.
A good software to start is xawtv.
Compile this program and execute it.
You have to select PAL and Composite-1.
Now you can see in a window the images of the input video signal.

There are also other software, some very interesting are mplayer and xine.
Both are used to see a video.
Furthermore mplayer packages contains also a program to compress a video using standard formats, for example MPEG4 or DIVX.
If you use also MS Windows there is a very good open source program to modify and compress a video: VirtualDUB.