Both above and over can also mean "more than something" :
The movie is suitable for children over/above 13 years old.
We've been having temperatures over/above 90 degrees Fahrenheit.
I value time over/above money.

Again, over is the more common choice. And there are a number of contexts in which above is not used at all, despite having a meaning that technically fits:
I've been waiting for over an hour. (NOT: I've been waiting for above an hour.)
Over 50 people attended. (NOT: Above 50 people attended.)

Both prepositions can also mean "more loudly and clearly than (another sound)," although again, over is more common:
I could barely hear the music over/above the noise of the crowd.

Each preposition also has many additional senses that are not shared by the other. See each word's entry for examples of those.

Below and Under

The prepositions below and under have similar meanings. Both can mean "lower than".
Cases where below is preferred -- Below can mean "at" or "to a lower level".

The people in the apartment below ours seldom go out.
Put your signature below.
I feel sick, I must go below. (= I must go downstairs to my cabin.)

Below: not directly under - We use below when one thing is not directly under another.

The sun sank below the horizon.
The climbers stopped 200m below the top of the mountain.

Below is used in measurements also where we think of a vertical scale.

The temperature is ten degrees below zero.
The Dead Sea is below sea level.
She is below average in intelligence.

Cases where under is preferred. We prefer under when something is covered by what is over it. Under is also used when things are touching each other.
There is something under the bed.
He wore a shirt under his sweater.

Under can mean "less than" or ˜younger than". Below is not used with this meaning.

There were under twenty people at the meeting. (= There were less than twenty people.)
You can't vote if you are under 18. (NOT You can't vote if you are below 18.)