I was reading some jokes on Facebook when I saw this. What does it mean and when will you use it?
Max has covered most of the answer well, so I will just offer an alternative interpretation of the final sentence.
As this is essentially speech, not prose, you can't assume proper use of grammar. An alternative reading of the final sentence is therefore "Fuck this, nigga.", where "this" refers to the instructions from "D", not "D" the person, and "nigga" is the person they are talking to. In this interpretation, it implies that the person saying this is going to paint the room blue despite D's advice.
In the interpretation proposed by Max, the sentence splits up in to "Fuck" and "this nigga". This is just an exclamation of contempt for "D" and doesn't necessarily imply any action from the speaker.
Further context may help you decide which meaning is more relevant. I would argue that the exact meaning as posted is ambiguous, but essentially amounts to the same thing: the speaker is saying that they are unhappy with being told they can't paint the room blue because Tamiya is a girl.
I would like to echo others by strongly advising that you don't use the word "nigga" as it is very likely to cause extreme offence to others. I would recommend never using swear words in general in any polite context, regardless of what other people are doing, and to only use them around others in an informal setting if they start using them first.
The following represents usage in AmE. I imagine there are similarities in other English speaking regions.
Usage alert Nigga is used mainly among African Americans, but also among other minorities and ethnicities, in a neutral or familiar way and as a friendly term of address. It is also common in rap music. However, nigga is taken to be extremely offensive when used by outsiders. Many people consider this word to be equally as offensive as nigger. The words nigger and nigga are pronounced alike in certain dialects, and so it has been claimed that they are one and the same word.
noun, Slang: Usually Disparaging and Offensive.
1. a term used to refer to or address a black person.
Origin of nigga
alteration of nigger, based on pronunciation
An exclamation of very strong defiance and contempt : Fuck you, friend, if that's your attitude (1940s+)
This nigga refers to D. D is someone's nickname. Overall, fuck this nigga is an expression of defiance and contempt, much like fuck you or fuck him. In this exact context, it would be equivalent to fuck him in terms of meaning (assuming D is a male).
In this instance, fuck this nigga is used instead of fuck him likely because of stylistic and cultural reasons. People who use nigga openly like this frequently use nigga instead of the usual pronouns in informal, casual speech. To their friends and people in their "circle", it's often meant as a friendly term. To people with whom they are not on friendly terms, it's again a term of contempt.
I would not recommend an English language learner to use these terms ever.
That text is written in African-American Vernacular English (AAVE). This is a dialect common all over the USA among African-Americans, particularly in areas where they are the majority (such as certain areas of most large cities, and in some rural inland areas in The South). Its famous internationally as the dialect the Blues, American Rap, and most Rock & Roll music is written in.
In that dialect, "nigga" is a term of (sometimes affectionate) annoyance a black person may use to refer to another black person, usually if they are seen to be either currently or perpetually behaving badly. The racial aspect is a very important point, as a white person using this same word to refer to a black person means something completely different (and quite offensive).*
"D" is possibly the person's first initial, but more likely slang shorthand for male genetalia, indicating this person is a female talking about a male she is in an intimate relationship with (and likely not married to).
Another aspect of AAVE is that cursing is usually sexual in nature (rather than religiously profane like it often is in wider English). So the last sentence is further emphasis of displeasure, and expressing an intent to ignore the guy's opinion.
I'd translate it into more stock English as:
My sexist boyfriend is insisting my daughter's room must not be painted blue, as he feels that is the incorrect color for her gender. I don't care what that jerk thinks.
* - Often some white people think they can use it in the more innocent sense. They are wrong. Context always matters.