Passive transport refers to movement of material across cell membranes wherein the material is moving from an area of high concentration to an area of lower concentration. This type of movement does not require energy.
An example: if there is more of Molecule A inside a cell than on the outside, Molecule A will want to move to the outside of the cell. In this way, Molecule A is said to be moving "down it's concentration gradient", from an area of high concentration to low.
There are three types of passive transport: simple diffusion, facilitated diffusion and osmosis.
On the other hand, active transport refers to the movement of material across cell membranes from an area of low concentration to an area of high concentration. This type of movement requires the use of energy, in the form of ATP. AS the material is moving "against it's gradient", it requires energy to occur.
An analogy: Say you are at a great house party. There are 100 people inside, standing shoulder to shoulder with you, and it's getting hot inside. You want some fresh air. Since the house is so crowded, it's easy to move yourself outside, where there are less people. You are moving "down the concentration gradient", from an area of high concentration (of people) to an area of low concentration. On the other hand, say you are standing OUTSIDE a great house party. You can see the crowd inside, and everyone looks like they are having fun. In order for you to get in, you would have to push inside, because it's so crowded with people. To push your way in, you would have to use some energy, because you would be mooing from an area of low concentration (outside) to an area of higher concentration (inside).