A minimap with features labled
Features of the Minimap
Minimaps were briefly introduced here. But let's go indepth into the information you can get from just glancing at the minimap once you understand what everything means.
Towers
First of all, you can see the positions of all towers that are currently standing. One a tower falls, its icon will be removed from the minimap completely. Downed inhibitors will still be shown, but they'll be greyed out rather than in color.
Wards and Traps
You can also see the positions of all of the teams' wards and other traps. As highlighted in the above image, there are two types of ward icons. Green wards (AKA stealth wards AKA sight wards) show up as three-pointed stars, and pink wards (AKA vision wards) show up as eyes. Ward icons are color-coded to the team that owns them, either blue or red.
Traps like 
Caitlyn's 
Yordle Snap Traps will show up as small dots, and like wards, they will be color-coded to the appropriate team.
Minions
Minions also show up as small dots, so you can see the positions of minion waves as lines or clumps on the map. Be on the look-out for large clumps of red or blue dots in front of an opposite-colored tower: this means that there are a lot of minions attacking that tower. If no champions defend it soon, the tower might fall, or at least a lot of gold and experience will be denied to the enemy team.
Champions
Perhaps most obviously, you can see the positions of all alive champions. Dead champions will not show up anywhere on the minimap. If at some point in time, you don't see as many champion icons as you expect you should, there are two possibilities: Either they're in the fountain, which can clip them off the corners, or there are two or more champions very close to each other somewhere on the map, which can cause only one portrait to show up on the minimap.
If a champion is currently recalling to base, or channeling the summoner spell 
Teleport, then a blue or red swirling circle will appear around the champion for the duration of that channel, as pictured above.
Jungle Camps
Finally, a minimap will show the location of jungle camps. Above, you can see the icon for blue wolves circled. All jungle camps except for Dragon and Baron have small gold icons; the icons for Dragon and Baron are slightly larger.
Current Main Screen Position
The current position of the main screen that you see is denoted by a white rectangle on the map. So if you're not sure exactly where on the map you're looking, the best thing to do is to look at the minimap and find the position of the white rectangle. You can also refer to these pages to know what part of the map the commentators are talking about.