A sentence fragment is an incomplete sentence. It is usually missing a subject or a predicate.
In the cornfield.
(missing subject)
The cows.
(missing predicate)
A run-on sentence is a sentence that has two or more ideas that are not connected properly. Most run-on sentences can be re-written into two separate sentences.
The sun is hot put on some sunblock.
The sun is hot. Remember to put on some sunblock.
I like to eat my dinner first I eat my dessert after.
I like to eat my dinner first. I eat my dessert after.