If passive transport is moving molecules across the cell membrane without using up energy, then logic follows that active transport is moving molecules across the membrane with the use of energy. This is used when a molecule needs to be moved against its concentration gradient. I'll put in the textbook definition again, which is "using energy to transport a particle through a membrane against a concentration gradient." This involves the use of membrane proteins called proton pumps, which are active transport channels. Proton pumps bring protons into the cell against their concentration gradient to build up a higher concentration of protons in the cell. The energy from the protons is then used to make ATP (adenosine triphosphate), a form of cell energy. ATP is used to power other kinds of active transport channels, like the sodium-potassium pump. The sodium-potassium pump brings potassium ions into the cell while releasing sodium ions from the cell in a way that is similar to facilitated diffusion (opening & closing channels). It builds up the amount of sodium outside the cell, and that sodium is later brought back in along with food molecules (like glucose) through another protein (these proteins working together are called coupled channels).
This shows a bunch of different types of transport through the membrane: active, passive and diffusion.
Citations (Web Sites)
Source of the first diagram (top of the page)
Source of the transport diagram