Active investing is a hands-on approach to investing that involves buying and selling to try to beat the market; passive investing is less reactive to the market and focuses on long-term gains. In other words, if you are an active investor you are watching the market closely and making regular changes. Passive investing is more of a hands-off approach, like with a 401(k) plan for which a management company makes most of the decisions for you.
Basically, through active monitoring, an active investor will try to predict the highs and lows of the market and then move your money around to maximize its growth. Because of this, active investing tends to have higher fees and higher risks. But, these risks also bring the possibility of higher rewards.
Passive investing is more focused on the long run, usually relying on the fact that the market (and thus your money) will trend upwards given enough time. Because there is significantly less active monitoring, passive investing tends to have lower fees. It also has less risk, but this sometimes comes with lower margins of growth.
Active and passive investing each brings different strengths and weaknesses to the table. To pick which one is right for your financial situation, start by evaluating your goals and time horizon.