A dangling modifier (or dangling participle) is an "-ing" or "-ed" verb form made to modify an absent or distant noun rather than, as it should, the subject of a sentence's main clause. Dangling modifiers are easy to spot if you ask yourself who or what is doing the action of the participial verb. Consider this sentence: "Relating this interpretation to the second stanza, a lovely image is formed." The subject of this sentence, "a lovely image," clearly does not do the work of "relating." That is the job of the reader, whom the passive construction inconveniently omits.
A subtler and consequently more common abuse of participles arises when writers employ them as an inadequate link between parts of a sentence:
Not only did the Wife of Bath have other company in youth, but she had five husbands, echoing the Samaritan woman.
What is the antecedent of "echoing"? In fact, the participle has no antecedent at all. It is not a modifier, but a tenuous link between the two parts of the sentence. A stronger connection is needed here - something like "but she had five husbands, like her predecessor the Samaritan woman."
Inadequate links are even more common with past participles. Participial phrases modify nouns, as in "Exhausted from their ordeal, the hostages refused to speak to the press." The adjective "Exhausted" clearly applies to "hostages." Contrast this with the example:
Based on early primary results, the senator decided to withdraw.
What noun does the participle "Based" modify? It does not modify "results" (the results are not based on something, rather something is based on the results). It also does not modify "senator" (we do not speak of people being "based on" things). In fact, the participial phrase here modifies a phantom noun, and you could repair this sentence by summoning up that phantom: "The senator's decision to withdraw was based on early primary results."
See WP 27; Purdue handout 24