Narrow/phonetic notation is usually written between square brackets and represents a specific pronunciation, such as from a specific recording.

Broad/phonemic notation is usually written between slashes and represents the underlying structure of the word, as stored in the mental lexicon.

The narrow/phonetic transcription can be determined based on the broad/phonemic transcription and the specific phonological rules in the language as well as various other factors including individual variation, pragmatic context (like speed of pronunciation and emotional state), and also possible free variation. It is impossible to truly transcribe at a precise phonetic level that captures all detail (that would simply be a recording) so narrow transcriptions can be at any level of relevant detail beyond the phonemes. Often a narrow transcription is based just on the 'normal' phonological processes in the language like, say, vowel reduction and T-flapping in English, and some obvious cases of assimilation, but it could be much narrower in other cases.

The broad/phonemic transcription can be determined from the narrow transcription by doing that process in reverse, but because the phonetic articulation can vary to theoretically an infinite degree and there are no constraints on how a word could be pronounced (or even mispronounced), it isn't possible to be entirely certain about the original underlying form. In short, you would need to recognize the word. This is the problem of (mechanical) speech recognition, as well as the field linguist converting recordings to a phonemic transcription. Once you recognize the word, then you can write it out phonemically. You don't even really need the narrow transcription to do that, after you know what the word is. If you very precisely know what all of the phonological rules in the language are, and if you have only phonological rules generating the narrow transcription (no individual variation, mistakes, etc.), then you could theoretically apply them in reverse to, in most cases, find just one underlying form that would generate that narrow transcription after the phonological rules apply. But that's a lot of ifs. It's easier to just recognize the word in context and go from there.

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So in answer to your second question, without any context and without even knowing the language it is in, I can't tell you what [igat] would be. I don't recognize the word in English. I can pronounce it (roughly ee-got), but I don't recognize it. Maybe it's in another language? Or maybe it's a phrase like an atypical (dialectal?) pronunciation of "He got".