The terms "pop-up waste" and "click-clack" are commonly used to describe the two different kinds of drain plugs found at the bottom of a sink basin or tub. In general, the purpose of drain plugs is to handle clogs or blockages that would otherwise compromise the efficiency of your drainage system. The difference between "pop-up" and "click-clack" waste is simply the way in which the drainage plug is opened and closed. The "pop-up" waste operates the drainage plug by lifting a lever, which is located directly behind the faucet, up and down. The "click-clack" waste doesn't involve any kind of lever system and is simply the drainage plug itself, which is operated by pushing it down (sometimes simultaneously turning it) in order to open and close the drainage plug. When the drainage plug pops up a "clack" is often heard and when pushed back down it makes a "click" sound, hence the origin of its name.
Both types of drain plugs can be purchased with "overflow" capabilities, which simply means notches are cut into the sides of the piece in order to allow water to drain straight through the plugs. This is beneficial if you have an overflow sink and accidentally leave the water on. Even if your drain plug is pushed down, with an overflow sink and overflow drain plug combination water will still be able to drain from your sink.
Below are visuals of the two different types of drain plugs and how they are operated. In the pop-up drain piece the overflow notches are visible. The first image seen here is the "pop-up" waste system. The vertical lever that's located behind the faucet and pulled up and down is attached to another horizontal lever, which directly connects to the drainage plug and enables it to "pop" up and down. There's no clear-cut advantage to purchasing ether the "pop-up" or the "click-clack" drainage plug.
Some people may prefer the look of not having the lever behind the faucet, and others may see the lever as a more convenient means of opening and closing the drain plug. Ultimately, it's a decision of personal preference, and sometimes it depends on the certain type of faucet you purchase and whether it has "click-clack" or "pop-up" capabilities.