There is a relationship between the focal length and the size of the sensor. For a larger sensor, the same focal length gives a smaller angle of view thus for an "equivalent" field of view you need to multiply the focal length by the same factor as the sensor size.
Before the onset of digital cameras, by far the most popular film format was commonly called "35mm film", which is technically "135 film". The film gets its name because it's the same width as 35mm movie film, but the images are shot horizontally allowing the images to be 36mm wide by 24mm tall on the film.
Because this film size was so common, we became most familiar with lens focal lengths as they applied to that frame size. On that frame size, focal lengths of 40-50mm were "normal", less than that eg 35mm were "wide angle" and more than that eg 70mm were "telephoto".
However, when you have a smaller film frame, or digital sensor, this changes.
Compact digital cameras have very tiny sensors, and accordingly they require much smaller lenses. On your compact camera, 7mm to 9mm is "normal", with 9mm on that camera having the equivalent field of view to 50mm on a "35mm" stills camera.
When you scale up the size of the sensor, the lens has to scale up too. This means heavier and larger lenses to reach the same angles of view. While it's relatively common for compact cameras to reach very high zoom levels (eg 14x) with a small lens, with a DSLR that would make the lens too large, heavy and expensive. If you wanted that amount of reach on a DSLR you would satisfy that by having multiple lenses and swapping them as needed. Typical normal zoom lenses for DSLRs have more modest zoom levels such as 28-70mm (which is only 2.5x).
A few side effects of scaling up the sensor, and along with it the lens, include:
- Larger, heavier lenses
- Better light sensitivity of the sensor
- Greater resolvability of really fine detail (larger pixels on the sensor relative to smallest resolvable detail of the lens)
- Narrower depth of field for same subject distance and framing, that gives a greater ability to blur the background.