A fragment is a portion of a sentence. It either lacks a subject or a predicate. That is, in and of itself, you do not have a fully expressed idea. Take, as an example: "Jack, a man of power in his hometown." This lacks a predicate and thus gives us no full idea. We have a mere subject with a description. Compare this with: "Jack, a man of power in his hometown, fell into abject poverty in his mid-30s." Here, we have a complete idea.
A run-on sentence occurs when you have two full sentences separated by no punctuation or conjunction. For example: "I kicked the dog the dog bit me." There are two complete ideas expressed here without punctuation or conjunctions to separate them. Depending on meaning, the two ideas may be presented as follows:
"I kicked the dog. The dog bit me."
"I kicked the dog because the dog bit me."
"I kicked the dog then the dog bit me."
"I kicked the dog; the dog bit me."
In each case, the separate ideas are separated, either by using punctuation or with the assistance of a meaningful conjunction that indicates cause and effect.
A splice (normally known as a "comma splice"), is similar to a run-on sentence in that it separates two independent clauses (i.e., sentences) with a comma, which is the incorrect punctuation for this situation. From the example above, "I bit the dog, the dog bit me" would be a comma splice.