What does "so there is only one ADC that is used by all analog pins" mean?
The chip can only read one analogue IP at a time because there is only a single ADC (A to D converter). You get 6 analogue IPs because there's a 6-way switch between the pins and the ADC. For most applications it makes no difference and it's all hidden by the Arduino environment.
Shouldn't the Mega do the job?
Yes, the UNO won't.
Rob