What is meant by 'resolving a vector into its components'?

So, let's think of a vector as an 'instruction' - a guide telling us to move in a specific direction for a certain distance. (Draw arbitrary vector A here). This is not the only way to move this distance away in this direction, however! We could also move to the left first, and then move to the point. (Draw two vectors which sum to A). To make things easier for us, we want to look at how far along we must go, and how far we up we must go (draw horizontal and vertical components of A). These are 'components' of A.
So, we know the directions of these components (horizontal and vertical), so now we just need to work out their magnitudes. Let's recall our trigonometry - as we're dealing with a right-angled triangle here - and think about Sin, Cos and Tan. Since we're looking at the opposite and adjacent sides of our triangle, we'll look at Sin and Cos. We know the hypotenuse of our triangle (it's just the magnitude of our original vector), so we can rearrange the trig functions to calculate the opposite and adjacent sides. Our angle is just the direction of the original vector. These horizontal and vertical vectors are the 'components' of our original vector A!