This is a pretty interesting scenario! Here's an interesting article that explains what would happen if Earth captured a second moon today.

This scenario though is obviously unrealistic today, and if a second moon was captured it would have occurred billions of years ago more realistically if it would ever happen. In this case, if the second moon was the same size as our moon, then the three body system would cause the two moons to collide and form a larger moon than our Moon. This would wreak havoc on the tides, seasons, and length of days. Life as we know it wouldn't exist, though life would adapt to the different conditions if they are suitable for complex organisms.

In another scenario where the second moon is rather small, more like the captured asteroid moons of Mars, not much would change, though it would be neat to have a second moon.