It doesn't sound like a consonant. Especially at the end of words. It sounds like a shortened vowel which it follows. English word "cat" for example [kʰæʔ]
It is a stop consonant, as it stops the airflow in the glottis.
The fact that you can't differentiate the sound doesn't mean it isn't a consonant - there are many, many, many consonants which sound indistinguishable if you don't have the right mother tongue. But they are distinguishable if you do speak the right language.
For example, semitic languages are languages in which the glottal stop is a consonant.
In some languages, glottal stop is associated with shortness of vowels, but so far as I know, this is not the case in English, except perhaps indirectly.
In English, vowels are shorter before a voiceless sound in the offset of a syllable, and also, the voiceless stops p t k are glottalized (said with glottal closure) in the offset of a syllable before a consonant. Then, in some circumstances, the glottalized p' t' k' lose their oral closure, leaving behind the glottal closure alone, which is a glottal stop. This loss of oral closure happens before a homorganic consonant (i.e., with the same place of articulation), or for t', before any consonant. These things happen only in some dialects.
So there is a relationship between the shortening of vowels and glottal stop (vowel is shortened before p t k, p t k are glottalized to p' t' k', which then turn into glottal stop), but it's coincidental.
In some other languages, glottal stop has more directly to do with vowel shortness. There is a language of Brazil, Shavante, which has "clipping" of word final vowels, which consists in shortening the vowels and appending glottal stop.
Some examples from Danish make it appear that glottal stop has something to do with vowel shortness: læser /ˈlɛːsʌ/ "reader", læser /ˈlɛˀsʌ/ "reads" are cited in the Wikipedia article on stød.
Although this is not so in English, in many languages, glottal stop is a phoneme -- just another consonant in the language's phonemic system. I'm not aware of any shortening of vowels in such cases.
Well, it is a consonant.
It is produced by stopping the airflow in the same way as in producing [p],[k] or [t], it is just made in different place. While the aforementioned obstruents are articulated by stopping the airflow respectively between lips (bilabial stop), root of the tongue and velum (velar stop) and tip of the tongue and alveolar ridge (alveolar stop), glottal stop is produced by stopping the airflow with glottis.
In English glottal stop function merely as an allophone of /t/ word finally, and (in some dialects) intervocally. But there are languages where glottal stop is a distinct phoneme. The example would be the Semitic family where it is written with letter aleph.