API Levels generally mean that as a programmer, you can communicate with the devices' built in functions and functionality. As the API level increases, functionality adds up (although some of it can get deprecated).
Choosing an API level for an application development should take at least two thing into account:
Current distribution — How many devices can actually support my application, if it was developed for API level 9, it cannot run on API level 8 and below, then "only" around 60% of devices can run it (true to the date this post was made).
Choosing a lower API level may support more devices but gain less functionality for your app. you may also work harder to achieve features you could've easily gained if you chose higher API level.
Android API levels can be divided to three main groups (not scientific, but what the heck):
Android 1.5 — 2.3 (API levels 3-10) — Android made specifically for smartphones.
Android 3.0 — 3.2 (API levels 11-13) — Honeycomb, Android made for tablets.
Android 4 (API levels 14-) — Ice Cream Sandwich — Current generation, a big merge with tons of additional functionality, totally revamped Android version, for both phone and tablets.