In English, "P" is called P and pronounced differently as if it was used in a word (usually, this is applicable for most letters in English).
Is there a concept of this in Korean? How would you refer to an independent "letter/character part" in a sentence in Korean?
For example, "ㅃ" is said as/called ____ and indicated this way in a sentence "___".
What a fun question! The consonants DO have names and can be used in spelling:
ㄱ = 기역 (giyeok)
ㄴ = 니은 (nieun)
ㄷ = 디귿 (digeut)
ㄹ = 리을 (rieul)
ㅁ = 미음 (mieum)
ㅂ = 비읍 (bieup)
ㅅ = 시옷 (siot)
ㅇ = 이응 (ieung)
ㅈ = 지읒 (jieut)
ㅊ = 치읓 (chieut)
ㅋ = 키읔 (kieuk)
ㅌ = 티읕 (tieut)
ㅍ = 피읖 (pieup)
ㅎ = 히읗 (hieut)
ㄲ = 쌍기역 (ssanggiyeok)
ㄸ = 쌍디귿 (ssangdigeut)
ㅃ = 쌍비읍 (ssangbieup)
ㅆ = 쌍시옷 (ssangsiot)
ㅉ = 쌍지읒 (ssangjieut)
The vowels do not have names; in spelling with them you just say the vowel sound as the vowel name.
Single consonants are given two-syllable names, mostly with two occurrences of the letter separated by 으. so ㅁ is named '미음', ㅈ is named '지읒', and so on. (As WEBjuju points out, there are some exceptions - see his answer!)
Double consonants are the same, prefixed with '쌍' - so "ㅃ" is called '쌍비읍'.
Vowels don't have special names, but in the case of the ones that are hard to distinguish, Korean speakers may 'spell out' the 'components' when speaking - e.g. 애 might be 'spelled out' verbally as '아-이'.
ㄷ = 디귿 (digeut) and ㅌ = 티읕 (tieut) - I'm not sure it completely follows the rules, there.