A syllabic consonant is a consonant that can form an entire syllable on its own, without any vowels. While these are usually not recognized in reading and writing, when transcribing speech phonetically we try to write out what is actually being said (and not what we think should be said). In these transcriptions syllabic consonants appear quite often. Even discounting onomatopoeia, we have three or four of them that commonly appear in English. While they aren't commonly recognized, we all use them.
M: Words like 'chasm' and 'prism' consist of two syllables, the second of which is simply an 'm'. Notice that this is different from words like 'anthem'. While 'anthem' could be pronounced with a syllabic 'm', it is commonly pronounced with a clearly audible schwa sound before the 'm'.
N: A little bit harder to describe than 'm', because while the syllabic 'n' certainly appears in English, it isn't written out as clearly as 'm' is. Words like 'kitten', 'mountain', and 'Dayton' are often pronounced with a syllabic 'n', at least in American English. Unfortunately I don't have examples of the syllabic 'n' in English English, but apparently it's also quite common in Japanese.
L: Again, not commonly written in English, but it may be the most common of the syllabic consonants. We often use it when pronouncing words ending in -le, like 'bobble', 'bundle', and 'bottle'. It can also be used when pronouncing other words, like 'legalize', and 'cardinal'. If you are trying to say these words out loud, don't try too hard to pronounce them 'correctly'. We normally speak in an informal way that tends to favor syllabic consonants. When speaking clearly we tend to add vowels.
R: 'R' is the reason why I said we had "three or four" syllabic consonants in English. If an 'r' appears after the schwa sound or the 'eh' (/ɛ/) sound some phoneticians will call 'r' a syllabic consonant, but others will call it a rhotic diphthong. (I prefer to call it a diphthong). If you do treat it as a syllabic consonant, you would transcribe words like 'her' and 'murder' without any vowels at all. It would also replace 'l' as the most common syllabic consonant.
If you are doing phonetic transcription using the IPA there is a way to indicate that an 'm', 'n', 'l', 'r', or any other consonant is acting as a syllabic consonant -- simply place a small, short vertical line underneath the letter in question. In practice this usually ends up being a small dot. In HTML you can make them by appending '̩' to the consonant you want to make syllabic; m̩, n̩, l̩, r̩. (This doesn't look right on my browser, placing the dots to the lower corner rather than beneath the letters. But it may work for you.)
Other languages use other consonants as syllabic consonants. In some cases these are used along with other consonants to form the syllable; all that is necessary for it to be considered a syllabic consonant is that it is the nucleus of the syllable.
HTML grabbed from http://www.phon.ucl.ac.uk/home/wells/ipa-unicode.htm#nonspac