IIR filters what does infinite mean?
To answer that question you have to know what "Impulse" and "Response" mean...

An "Impulse" is a simple pulse.  Digitally it would be a sample with a maximum value, while all of the other samples before and after it would be zero.  If you listened to this, you would hear something like a pop or firecracker.  

The "Response" is the output of a filter (or something else) is given the impulse.  

For example, you can listen to the "impulse response of a room" by going into the room, doing a simple hand clap, and listening to the echo.  It would take some practice to get the hand clap to be as "sharp" as possible.   Getting the impulse response of a filter is the same way but instead of a hand clap you use a simple pulse and instead of a room you have a filter.

If you look at the impulse response of a filter, or a room, you will see the output wiggle for a time after the impulse (and sometimes it wiggles before, too).  In the room you heard this wiggle as the echo.  In a filter this wiggling is directly related to the frequency and phase response of the filter.  In a room, the amount of time you hear the echo is called the "reverb time"-- there is no corresponding term for a filter, but it is part of the impulse response.

Now, a FIR filter (Finite Impulse Response) is finite because the impulse response time is limited by the math.  It's mathematically impossible for the impulse response time to extend beyond the number of taps in the filter-- therefore it is finite.

An IIR filter, on the other hand, does not have this mathematical limitation in the impulse response time.  If given an infinite mathematical precision, an IIR filter can wiggle the outputs forever.  Of course, in a practical sense it never goes on forever since at some point the wiggle gets smaller than the precision of the math used and so goes away.