long and short of lenses
A lens is described by its focal length and its minimum aperture. The focal length is how far from the lens to where the converged light comes into focus. The focus point will be on the digital sensor or photographic film.
The minimum aperture is the largest opening the physical blades can be in terms of f-stops. It is calculated by , where f is the focal length, and D is the diameter of the lens. A large aperture (big opening) has a small number. The smaller the f-stop, the more light the lens lets onto the sensor/film. For example, a 50 mm f/2.8 lens is a commonly specced lens that has a large opening and thus lets in a lot of light. Using the formula above, this means that the aperture itself has a diameter around 17.8 mm (50/2.8= ~17.8).
Because of geometry and cost of manufacturing, it is rare to find low aperture lens with a long or variable focal length. Typically a lens with a focal length of 200 mm might have a minimum aperture of f/3.5. There's a trade off from the benefits of the longer lens with capturing less light.
What are these benefits of a longer lens you speak of?
A telephoto lens is generally agreed to be between 100-300 mm focal lengths. The prefix tele- is the same as telescope, which tells you it makes far objects appear closer. Because the subject is closer, there will be less in the peripheral, the image will be cropped smaller but the subject appear larger relative to the edges of the frame. The subject will be zoomed in. Not to be confused with a zoom lens, but first what about a shorter focal length?
A short focal length has the opposite effect, it will be a wide angle view. Wide lens are generally those less than 50 mm which is considered "normal". 50 mm is similar to the focal length of the human eye and so when viewing images taken with 50 mm focal length they appear "natural". A common wide angle lens is around 28 mm. The smaller you go the more distorted the image will become and you get the fisheye effect (around 18 mm).
If you keep the subject filling the frame you can visualize the distortion very clearly like this. In portraits, this distortion is actually very useful. If your subject is sensitive about the size of their nose for example, you can use a longer lens to compress that feature of their face. Notice in the animation how their nose becomes less exaggerated with the longer focal length. This is also an example of "zoom with your feet", you want the distortion of the telephoto but now you're looking up their nose instead of a nice portrait, you need to increase your distance to the subject to fill the frame appropriately.
Other misc things.
A zoom lens has a variable focal length, it might be able to change from 35-135 mm, but will have a much higher f-number typically f/3.5 to f/5.6. So there's a tradeoff between carrying a single lens that can cover wide and telephoto (you might struggle in low light). Or you might use a 'prime' lens (which just means its a fixed focal length), but will have a smaller f-number (larger opening) and do better in low light.
A 'fast 50' is a 50 mm lens with a low f-number, f/1.7 or f/1.4 for example. These are very versatile and used in many situations because they offer low light capabilities and little distortion (relative to how our eyes see the world).