A simple sentence is the most basic of all sentences, containing one and only one clause. A simple sentence will always have a subject (what the sentence is about) as well as a predicate (explaining what is happening to the subject). Beware though, that the simplest of sentences may have an understood subject. These generally take the form of a command with the understood subject being "you."
Run!
|
(You) | Run
-------------
sub. | pred.
So long as the sentence contains only one clause it may be much longer than this first example.
Susie, run!
|
Susie | run
-----------------
Susie, run away from the lion!
|
Susie | run
-----------------
\
\ away
| \
| \ lion
from ---------
\
\ the
\
Susie, run away from the great, big, scary lion before he eats you!
|
Susie | run
-----------------------
\ \
. \ away
\ | \
. | \ lion
\ from ---------------------
. \ \ \ \
\before \the \great \big \scary
. \ \ \ \
\
. |
\ he | eats | you
--------------------------
Simple sentences, in the evolutionary chain of written communication, are a rung in the ladder above clauses which themselves are the parents of phrases. Additionally, simple sentences are often the first successful phrases a child will put together when learning language. Direct and poignant, simple sentences work admirably as summation or conclusion points; their overuse however may have an overall "simpleton" effect on your work.
For further edification, perhaps a perusal of clause to review where sentences come from is in order. Or rather we should forge ahead and learn about compound sentences...