A misplaced modifier is attached wrongly to the subject noun or the verb, often because it refers to the first noun or pronoun:
"Arriving late, I was able to catch the train after all". Here "arriving" modifies the subject "I" rather than the object "train" and so clearly the intended meaning is "I caught the train although I was late". To satisfy people who dislike these types of participle in principle, it is best to rewrite the sentence like this: "Although I arrived late, I was able to catch the train after all". Of course, if the intended meaning had been "I was able to catch the train after all because it was late." then "Arriving late, " in the original sentence would have been a misplaced modifier and it would have been essential to rewrite the sentence to avoid confusion.
Not all dangling modifiers cause such confusion. For instance "Being ill in bed, the telephone startled me when it rang" is relatively clear in its meaning, since inanimate objects cannot be ill. Compare it with "Being ill in bed, the telephonist startled me when she rang", in which it is impossible to tell whether the telephonist or the speaker is ill.
Either sentence can be rewritten in several different ways but there is only a semantic need to rewrite the second one: "Being ill in bed, I was startled when the telephonist rang" or "While I was ill in bed, the telephonist startled me when she rang" or even "Ringing suddenly, the telephonist startled me while I was ill in bed".
As may be seen, the dangling modifier is often a participle which has been placed in the wrong position in relation to the subject/object of the sentence (or the subject/object to the participle). For this reason it is also called a dangling participle.
However, it is not only participles that may be misplaced. Adverbial phrases or adverbs may also be misapplied in this way. One of the greatest stylistic controversies of the last thirty or so years has been that of the dangling adverb "hopefully". Observers began to object when they first encountered constructions such as "hopefully, the sun will be shining tomorrow", arguing that it is not the sun but the speaker who hopes that the sun will shine. However, in recent years this usage has become much more acceptable to many, probably due to the fact that its semantics is reminiscent of the German "hoffentlich - it is to be hoped that" which does, in fact, show that it is the speaker who "hopes that the sun will shine".