A murmured sound is a phoneme produced with breathy voice. In fact, it is very common to call these phonemes 'breathy-voiced', for example, 'breathy-voiced ɦ'. However, breathy voice is most often used in English-speaking countries to refer to a sometimes-pathological voice quality that is used across all phonemes. This vocal quality is perhaps most familiar as the 'Marilyn Monroe voice', and is not generally seen as a pathology by most people, some of whom may very well cultivate it. A voice therapist might disagree, as it is an indicator that the vocal folds are not closing fully, perhaps due to polyps or vocal nodules.
However, a murmured sound is not a sign of glottal pathology, but one way that some languages form sounds. In English we do not have many murmured sounds, but we do regularly use a murmured h, usually written using the IPA symbol [ɦ]. In most words we use a voiceless [h], in which the vocal folds are open; however, in the middle of words, and most particularly in between two vowels, the [h] is formed with the folds vibrating but slightly apart. This allophone is sometimes referred to as 'voiced h', although it is not truly voiced. It is most properly referred to as 'murmured h'.
With practice, you should be able to recognize the difference between the [h] in 'heart' and the [ɦ] in 'aha' or 'ahead'. With a good bit of practice, you may be able to produce them in isolation, and prolong them. You will find that it is harder to hold [h] for long periods of time, as the air escapes more quickly when the glottis is fully open. On the other hand, not only will you be able to hold [ɦ] for longer, but you will also be able to alter your pitch while holding it, as the near-closure of the glottis does allow for some voicing.
In other languages there are any number of murmured consonants and vowels; some of these have unique symbols, but in general the murmur quality is indicated by putting two dots under the IPA symbol, for example, a̤ and d̤. At other times murmured phonemes may be indicated by a superscripted ʱ. (When you wish to indicate a breathy voice quality for a sentence or an entire section of a transcription, you proceed that portion of the transcript with a capital V with the two dots underneath).
You may find references to 'murmured stops'; in this case a stop is followed by a murmured consonant/vowel combination, often [ɦ] with a vowel. This is common in Hindi and other Indian languages. This is also indicated by a superscripted ɦ.
When writing murmured sounds from the perspective of voice science, it is usual to use square brackets rather than the slashes usually used in the IPA ([ɦ] rather than /ɦ/); this is because the IPA has a special set of symbols for voice quality, called the "Voice Quality Symbols", often referred to as the VoQS. In this writeup I have been writing from the perspective phonology, but have used the square brackets because they are used in the IPA to denote an allophone, as opposed to a phoneme. Had I been using Hindi words for my examples, I would have used the slashes.