As it's said almost everywhere, "ㄱ" is pronounced with a G sound and transliterated as G. However, it's very common to come across situations where ㄱ is romanized as K, specially in names, like Kim, instead of "Gim".
But why? When to use K or G?
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Romanization of ㄱ is ambiguous because its aspiration is closer to English 'k', but its voicing is closer to English 'g'. The official Romanization is 'g', but exceptions are made for certain names.
According to The Sounds of Korean by Miho Cho and William O'Grady, pg 39:
In terms of aspiration, Korean ㄱ falls closer to English unaspirated 'g' than to moderately aspirated 'k'. But in terms of voicing, it is closer to 'k' because both are voiceless. As a result, there is confusion when Korean words are written in English, which is why there is uncertainty over how to spell 김치, which is sometimes written as kimchi and sometimes written as gimchi. The current official romanization system writes ㄱ as g at the beginning of a word.
Word initial it is unvoiced, hence [k]. In medial positions, it becomes voiced, hence [g]. This is a regular phonological process, so native speakers without linguistic training are typically unaware of the difference. Also note that is a regular pattern found in many other Korean consonants.
How you wish to romanize it depends on the romanization scheme. There are several to choose from.
Simplistically speaking, it's neither English exactly the same as English 'G' or 'K', but (to my ears at least) somewhere in between. The Revised Romanization of Korean suggests G for initial letters, and K for final.
The older McCune–Reischauer system Romanised ㄱ as K, and the 'K' has stuck for very common words such as the name 'Kim'.