Lagging metrics are usually the ones we care about most. In business they're things like revenue, sales, or profit. The trouble is, we can't act always act on those directly. By the time we assess our revenue everything we did to get there has already happened, so if we don't like it, it's too late.
Leading metrics are what we need to focus on to drive the lagging metrics. If we want to lose weight we might be better to focus on eating less, and eating healthier, working out more, and sticking to our workout schedule. To increase our revenue we may want to look at deal size, ease of sign up, or our referral rate.
Paying attention to, and improving the leading metrics gives us our leverage over the lagging metrics we may really care about.
Also see: Analytics maturity