The simple version:
A fragment is a string of words that is an incomplete sentence. Usually this is because it is missing a verb.
A phrase is a group of words that can be used in a sentence.
The complicated version:
A sentence fragment (or simply "fragment") is an incomplete part of a sentence. If someone says that a string of words is a "sentence fragment" rather than a sentence, they mean that it is not a grammatically complete sentence.
In everyday speech, "phrase" can refer to any group of words. In linguistics, a phrase is a group of words that form a constituent - a single unit in the syntax of the sentence.
This means that all phrases can be considered fragments, but not all fragments are phrases.
For example:
"My black cat is happy" is a sentence.
"My black cat" is a phrase, and is also a fragment.
"black cat is" is a frament, but not a phrase (it is neither a complete noun phrase, or verb phrase)