Note: This is one of those sections that requires basic Algebra. The next non-Algebra section is Moments, Not What You Thought They Were.
A vector is a mathematical way to represent a force as the sum of its components (see the Parallelogram Law for an explanation). To be consistent, and for it to work correctly, the two components are always placed parallel to the x and y axes (one's parallel to the x-axis, the other to the y-axis) in a coordinate plane (like the horizontal and vertical lines on graph paper). Now we need to explain what this means. One of the component forces is Ai, the other is Bj. The two letters i and j are just little tags to represent in which direction the force is going. The x-direction (horizontal) is represented by i, the y-direction (vertical) by j. The other two letters, "A" and "B", are the magnitudes of the two component forces. So, to rephrase, we've broken our original force down into two different forces, one is completely horizontal and has magnitude "A" and the other is completely vertical and has magnitude "B". [Example 1] If the force is almost flat, like in the example at right, "A" will be large as compared to "B", since more of the original force's energy is directed in the x-direction. However, neither "A" nor "B" can be larger than the magnitude of the original force. This can be proved by looking at a parallelogram (or, since the x and y directions are perpendicular, looking at a rectangle would be better). You'll notice that none of the sides are longer than the diagonal.

Now, just to confuse you one more time, there's another kind of vector. It's called a unit vector and is used purely to represent the direction of the force (the magnitude isn't taken into account). You can usually recognize them because both "A" and "B" are less than or equal to 1 (that's where "unit" vector comes from, like in the "unit's digit"). As a follow-up to the previous paragraph, the closer "A", or "B", is to 1, the more the force is pointing in the i, or j, direction respectively. One thing to note is that values of "A" and "B" in a unit vector correspond to each other and you can't just make them up. You either have to figure them out yourself (see next section) or have them given to you (in which case someone else figured them out).

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