The technical terms and colloquial usage are a bit different here:

• Focal length is a technical characteristic of a lens that directly influences its angle of view, i.e. how much of the entire field of view it will project onto the sensor. Large focal length results in a small angle of view, which means the image contains less, but it's shown larger.
• Magnification is the ratio between an object's real size and the size at which a lens can project it onto the sensor. This is a function of focal length and the closest distance at which the lens can focus. Most lenses have a magnification of 0.3 or smaller: i.e., they cannot show objects at more than 30% of their real size on the sensor. Macro lenses are defined as having a magnification of 1 or larger, which they achieve by being able to focus more closely than regular lenses.
• Zoom is the ratio between the smallest and largest focal lengths in a lens where it's variable. It's often interpreted as "how much larger than the naked eye can the lens show something," but that is only true if the lower end of the focal length range is equivalent to the angle of view that the naked eye has, which is equivalent to what a full frame camera shows with a 50mm lens. There are wide angle zooms that show things "smaller" than the naked eye even at the upper end of their 2x zoom range. So the zoom factor by itself is meaningless without knowing where it starts.
• Crop factor is the ratio of the sensor's size to a full frame sensor that has the "standard" 35mm size. This is important because a smaller sensor shows a smaller part of the image the lens projects, which has the same effect in many ways as using a larger focal length. A common sensor size is APS-C, which is 1.5 times smaller than full frame, which means that a 50mm lens on an APS-C sensor will result in the same angle of view as a 75mm lens on a full frame sensor. Which is why focal lengths are not directly comparable between cameras and you'll often hear people talk about "35mm-equivalent focal length" when computing angles of view. It's also the reason why compact cameras can be so compact: they have much smaller sensors, which means that for the same angle of view they can use a lens with a much smaller focal length, which means the lens itself can be much smaller (and is cheaper to make).