For the AtMega328's ADC the datasheet says
• Up to 76.9kSPS (Up to 15kSPS at Maximum Resolution)
So, sampling two channels at 10bit resolution and 44.1kHz is not possible.
If 8bit resolution is enough, and if there's only one channel to sample, it should be able to do the job.
To answer Connor's comment:
Of course, this does only apply when using the internal ADC. Connecting an external ADC (via SPI or whatever) makes it possible to sample data at the desired speed.
But anyway, the next question is: Is 16MHz fast enough to process the sampled data (which are received every (22.6/channels) microseconds? ;-)